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INDUCTIVE vs. DEDUCTIVE METHODS 
OF TEACHING 



!E&ufatt0ttal ^Bgrliologg Monograyl|g 

No. 11 

Inductive versus Deductive 

Methods of Teacliing: An 

Experimental Researcli 

W: H. WINCH 

External Member of the Psychological Board of Studies for the University of London 

Chairman of the Committee of the Teachers' Guild of Great Britain and Ireland 

on Psychological Research in Schools ; Lecturer for the London County 

Council on Pedagogical Methods in Schools. 

Author of "Problems in Education,'^ ''German Schools, ^^ "When 
Should a Child Begin School, ' ' etc. 




laltimorp. 1. &. A. 

WARWICK & YORK, Inc. 
1913 



.V\i5 



Copyriehl, 1913 
WARWICK & YORK. Inc. 






EDITOR'S PREFACE. 

It affords me great pleasure to call editorial atten- 
tion to this interesting and instructive contribution 
to experimental pedagogy. Mr. Winch writes with 
the authority of long experience born of his profes- 
sional duties as one of the official inspectors of Eng- 
lish schools. He is, indeed, well known as the first 
Englishman to bring the technique of experimental 
and statistical methods to bear upon the actual prac- 
tical problems of the school. 

Those who have followed with any care the modern 
developments of educational theory know how sig- 
nificant is that trend of investigation which seeks to 
study the concrete problems of education at first 
hand in the classroom and with all the exactness of 
experimental control. The movement for experi- 
mental pedagogy is yet in its infancy, but it has 
already shown the possibilities that lie before it. In 
the Journal of Educational Psychology, with which 
this series of Educational Psychology Monographs 
is affiliated, there has appeared of late an important 
series of articles which show for various school sub- 
jects what important problems offer hope of solution 
by experimental investigation. This monograph 
presents what is at the very least a first approxima- 



Z INDUCTIVE VS. DEDUCTIVE METHODS. 

tion toward the solution of one of these vexed ques- 
tions of educational practice : Is it better to follow 
deductive or inductive methods in the teaching of 
various types of subject-matter 1 The presentation 
has the special merit of being sufficiently detailed 
that any teacher who desires to do so may of himself 
repeat the experiments and verify the conclusions. 

G. M. W. 



AUTHOR'S PREFACE. 

This is, I believe, the first attempt to decide be- 
tween the conflicting claims of 'inductive' and 'de- 
ductive ' methods by experimental procedure. In the 
'world of science' it is not usual for results to be 
accepted unless the methods by which they have been 
obtained are described in such detail as enables other 
workers to repeat, corroborate, or modify them. Nor 
are they regarded as valid unless they are obtainable 
under widely differing external circumstances. To 
produce similar evidence for educational science will 
be the aim of all serious workers in education during 
the next two or three decades, and I am therefore 
offering this research as a contribution to the scien- 
tific knowledge of the results of inductive and de- 
ductive methods in actual application under school 
conditions. 

I am quite well aware that much valuable know- 
ledge is collected by school administrators and school 
inspectors during the ordinary course of their work. 
They know much about the results of the application 
of different methods in different schools. But to dis- 
entangle all the contributory factors — even to realize 
them — is very difficult, and inspectors are likely to be 
misled ; for the teacher is, naturally, mainly desirous 
of showing that his school is a good one, and not of 



4 INDUCTIVE VS. DEDUCTIVE METHODS. 

settling, by experimental tests, tlie value of a par- 
ticular method. The work reported in this mono- 
graph is not subject to this source of error, since the 
teachers, in this case, were working luitJi the experi- 
menter, and not against him. It is my firm and ever- 
growing conviction that without that kind of co-op- 
eration on the part of teachers there can never be, in 
an applicable sense, a 'Science' of Education. 

W. H. W. 
London, September, 1912. 



CONTENTS. 

statistical Note 7 

I. Introduction 11 

II. The Problem of the Experiments 17 

III. The General Plan of the Experiments 20 

IV. First Series of Experiments : 

1. General Characteristics of the Children Who 

Worked the Exercises 2.3 

2. The Preliminary Tests 23 

3. The Method of Marking the Preliminary Tests . 25 

4. The Teaching of the TSvo Groups 31 

5. The Immediate Testing of the Two Groups ... 35 

6. The Marking of the Tests 30 

7. The Subsequent Testing of the Two Groups on the 

Same Subject-matter 37 

8. The Testing of the Two Groups on New Material . 38 

9. The Marking of the New Material 40 

10. Chronology of the Experiment 44 

11. Results : 

(a) The Marks for the Preliminary Tests ... 45 

(b) The Marks for the Test Immediately After 

the Definitions Had Been Taught and 
Learnt 4fl 

(c) The Marks for the Tests of Deferred Repro- 

duction 47 

(d) Correlation Between Immediate and De- 

ferred Reproduction 49 

(e) Results When the Two Groups Are Tested on 

New Material 50 

12. Pedagogical Conclusions 53 

V. Second Series of Experiments : 

1. General Plan 55 

2. The Preliminary Tests and the Method of Marking. 50 

3. Chronology of the Experiment 59 

4. The Final Tests and the Method of Marking . . 60 

5. Results of the Experiment : 

(a) Results of the Preliminary Tests .... 61 

(b) Results of the Tests in Immediate and De- 

ferred Re]iroduction 63 

5 



(c) Correlation Between Immediate and De- 

ferred Reproduction 65 

(d) Results of the Test on New Material ... 67 
VI, Third Series of Experiments : 

1. General Plan 69 

2. The Preliminary Tests and the Method of Marking. 70 

3. Chronology of the Experiment 75 

4. The Final Tests and the Method of Marking . . 76 

5. Results of the Experiments : 

(a) Results of the Preliminary Tests .... 90 

(b) Results of the Tests in Immediate and De- 

ferred Reproduction 92 

(c) Correlation Between the Results of Immedi- 

ate and Deferred Reproduction .... 95 

(d) Results of the Test on New Material ... 96 
VII. Fourth Series of Experiments: 

1. General Plan 100 

2. The Preliminary Tests and the Method of Marking. 101 

3. Chronology of the Experiment 104 

4. The Tests of Immediate and Deferred Reproduc- 

tion 107 

5. The Test of Application to New Material .... 107 

6. Results 114 

VIII. Fifth Series of Experiments: 

1. General Plan 119 

2. The Preliminary Tests and the Method of Marking. 120 

3. Chronology of the Experiment 122 

4. The Tests of Reproduction 124 

5. The Test of Application to New Material .... 129 

6. Results : 

(a) Of the Preliminary Tests 133 

(b) Of Immediate Reproduction 134 

(c) Correspondence Between Immediate and De- 

ferred Reproduction 135 

(d) Results of the Test on New Material . . .138 
IX. General Summary 140 



STATISTICAL NOTE. 

Suppose we have two measurements of any mental 
function for a number of children, that the second 
measurement gives higher results than the first in 
most cases, and that the average mark for the second 
measurement is a little higher than for the first. 
May we, therefore, conclude that some general tend- 
ency is at work, or must we regard the higher aver- 
age of the second measurement as the result of 
'chance' or mere variability? To answer this ques- 
tion I propose to illustrate the usual statistical check 
on results of this kind by means of one or two exam- 
ples. Suppose the children are measured for their 
power of spontaneous definition ; that, a week later, 
they are measured again, and that the marks are as 
shown in the following table : 

First Second 
Name. measurement. measurement 

A. B 9 10 

CD 8 9 

E. F 7 8 

G. H 6 7 

I. J 5 6 

K. L 4 5 

M. N 3 4 

O. P 2 3 

R. S 1 2 

Average, 5 Average, 6 

Common sense has no difficulty in deciding that 
there is a 'general tendency' to imj^rovement from 
one exercise to another. Let us now calculate the 



8 INDtrCTIVE vs. DEDUCTIVE METHODS. 

'probable errors.' The 'probable error' of an aver- 
age is i , where 'a' is the standard deviation, 

Vn 

and *n' is the number of cases measured.* Worked 
out on this formula, the 'probable error' of the aver- 
age 5 is approximately .6, and of the average 6 is also 
approximately .6. The 'probable error' of the dif- 

ference hetiveen two averages is .67449 V^^-^ ' 

n 

where '^i' is the standard deviation of the first aver- 
age, 'o-o' is the standard deviation of the second aver- 
age, and 'n' is the number of cases measured. Ap- 
plying this formula to the present example, we have 
the 'probable error' of the difference between the 

+ arj^Ao /(2.6)^4- (2.6)^ ,., . 

two averages — .6/449 V j — -— —, which is 

n 

approximately .8. 

It is required statistically that the difference be- 
tween two means shall be twice (or more) the 'prob- 
able error' of that difference before the difference is 
supposed to be 'significant,' that is, indicative of a 
general tendency. But the difference between the 
means in this case is only 1 and its 'probable error' 
is .8, so that, apparently, we have no 'significant' 
difference at all. 

But let us consider one more illustration in which 
the averages are the same, but in which common 
sense would not find a general tendency to improve- 
ment: 



♦Simple illustrations, in which V is found from easy examples, 
are given in the statistical note attached to my monograph, When 
Should a Child Begin School? 



STATISTICAL NOTE. 





First 


Second 


Name. 


measurement. 


measurement. 


A. B 


9 


2 


C. D 


8 


3 


E. F 


7 


4 


G. H 


6 


5 


I. J 


5 


6 


K. L 


4 


7 


M. N 


3 


8 


O. P 


2 


9 


R. S 


1 


10 




Average, 5 


Average, 6 



The difference between the means is again equal 
to 1, and the 'probable error' of the difference, cal- 
culated just as before, is .8. Statistically, therefore, 
we are precisely in the same position as in the pre- 
vious example, and there is no 'general tendency' to 
improvement. But quite obviously the two cases are 
by no means similar and their ' probable errors ' are 
not the same, for we have overlooked the positive 
correspondence between the first and second meas- 
urements of A. B., C. D., and the rest in the first 
illustrative case and the negative correspondence in 
the second illustrative case. The theory of statistics 
takes account of this correspondence, or lack of cor- 
respondence, in the following formula for the 'prob- 
able error ' of the difference between two averages : 



.67449 V "'- + "^^~^^"^^% 
n 

where ' (Ti ' is the standard deviation of the first aver- 
age, 'o-o' is the standard deviation of the second aver- 
age, 'r' is the coefficient of correlation between the 
two series of measurements, and 'n' is the number of 
cases measured. I suggest now that the correlation 



10 INDUCTIVE VS. DEDUCTIVE METHODS. 

coefficient be worked out for both cases. In the first 
case 'r' will be found to be + 1 and in the second case 
— 1.* Let us then apply the formula, corrected for 
correlation, to the two illustrative cases. The 'prob- 
able error' in the first case is 

.67449 y^ (2.6)''+(2.6)'--2(+l)2.6~X2:6 

n 

It will be seen at once that the expression disap- 
pears, for 2 (2.6) (2.6) = (2.6)^ + (2.6) = : that is to 
say, the difference between the averages is perfectly 
representative of the two series of measurements, as 
common sense would suppose. In the second case 
the 'probable error' is 

.67449 / (^-6)- + (2.6)- - 2 (- 1) 2.6'xT6 ^ 

n 

that is, double what it was when the negative corre- 
lation was neglected. It now reaches 1.6, and is 
greater than the difference between the averages, 
which is only 1. Hence the conclusion is against any 
general tendency, again in accordance with common 
sense. 

These illustrations will probably be sufficient to 
show that the use of probable error formulae without 
regard to correlation may be very misleading, and 
also that mere averages, without some indication of 
the nature and extent of the variability of the meas- 
urements, may be even more so. 



*Easy illustrations will be found in the statistical note previously 
referred to. 



INDUCTIVE VERSUS DEDUCTIVE METHODS OF 
TEACHING: AN EXPERIMENTAL RESEARCH. 

I. INTRODUCTION. 

In England — it is for Americans to speak for their 
own country — there is a widely-spread opinion that 
the theory and the practice of teaching are two very 
different things. The young student leaves the nor- 
mal school or training college, and, doubtless crudely 
enough, begins to put, or to try to put, into practice 
some of the pedagogical methods which he has been 
taught as theoretically sound. 

Not infrequently — I had almost said invariably — 
his confidence in his theoretical instruction receives 
a violent shock. His superiors in the school assure 
him that he will do no good if he goes on like that. 
"What is worse and much more disconcerting (for, 
after all, principals and head masters must find 
fault; it is their metier), his confreres look on with 
amused tolerance and 'chaff' him about his 'new- 
fangled' ways. Then, dropping into friendly confi- 
dence, they explain to him that it was all very well 
for him to 'get up' and describe methods of that sort 
in examination papers; it was expected of him, and, 
naturally enough, he wished to get his certificate of 

11 



12 INDUCTIVE VS. DEDUCTIVE METHODS. 

proficiency and to do credit to liis college. Exam- 
iners required these things; they were unpractical 
persons like professors, but, of course, a wise student 
humored them; besides, how else could he pass his 
examinations ? Let these fellows take off their coats 
and come and do a day's teaching in the schools, and 
they would very soon change their opinions. Their 
stuff is all theory, and in actual school life is simply 
no good. Now you have become a man, you must put 
away from you childish things ; and so on. Thus the 
experienced and disillusioned confreres to the neo- 
phyte. 

It is not clear that taking off their coats would 
assist much in such professional conversions as are 
here anticipated, but the suggestion is a protest 
against what the teachers regard as a rather vision- 
ary and unpractical existence. 

If this rude shock resulted in complete divorce, 
there would be some hopes of other and happier mar- 
riages between theory and practice later on ; but, in 
England at least, what happens is rather of the na- 
ture of a judicial separation. 

The theoretical methods are not absolutely dis- 
carded; they are laid by and put in evidence only on 
special occasions ; the practical methods do duty day 
by day. For it is dangerous, from the standpoint of 
professional advancement, for the teacher boldly to 
renounce the methods he has been taught ; he is pur- 
sued all his life by unpractical, theoretical persons, 
to wit, inspectors, and he will often deem it to his 
advantage to profess a method he does not believe in. 

Head masters, too, mindful of the repute of their 



INTRODUCTION. 13 

schools, will say, "Yes, that's all right, but don't do 

that when Mr. I r comes ; he does not like it ; he 

thinks 5^011 ought to teach that this way." 

Well, yes, no doubt, but what is there in all this 
but the usual difficulty which besets the young in- 
structed person when he takes an actual place in the 
working world : it is the old difficulty of science ver- 
sus practice. In a few years the teacher, like other 
people, will have allowed his theory and his practice 
to interpenetrate, and both will have been improved. 
In such wise may we suppose an experienced admin- 
istrator pooh-poohing my criticism. 

If I could admit this, my complaint would indeed 
lose much of its poigancy. But I do not admit it. On 
the contrary, I believe that with the great majority 
of teachers there remains permanently an irrecon- 
cilable breach between dominant theory and current 
practice. It is true that experienced teachers^ — 
some of them — will attend lectures about educational 
topics. Two men speaking somewhat loudly after 
leaving a lecture hall — modesty forbids me to name 
the lecturer — said one to the other: "I didn't get 
much from him ; he 's like all the rest of 'em. " " Oh ! 
I don't know," said the other, with a give-the-devil- 
his-due air, "one gets ideas." "Yes," was the 
prompt reply, "but they don't work." And this, let 
us quite clearly understand that, is not merely an 
expression of a private grumble ; it is a strongly 
held and often a clearly reasoned view. 

There are always "new methods" in education, of 
course, and I hope that those of us who hold the very 
newest of them are more or less prepared to see our 
darlings cold-shouldered for a newer birth. Still, 
behind all temporary fluctuations, there is a line of 



14 INDUCTIVE VS. DEDUCTIVE METHODS. 

steady meaning in such phrases as 'new method,' 
'inductive procedure,' 'developmental method,' 'psy- 
chological method,' et id omne genus. And behind 
all temporaiy oscillations there is a steady trend of 
opinion amongst experienced teachers that these 
methods have certain serious disadvantages; that 
though they may be valuable for show j^urposes in 
teaching, they are too slow, and the information thus 
acquired is not really available when it is wanted. 

An experienced head master in London wrote to a 
lecturer on pedagogy in the following terms : 

"I (recently) asked a question on the difficulty of 
covering a present average course (by using the new 
methods) in the time given to it on the school time- 
table, and I should like to press the point and illus- 
trate its importance from my knowledge of the views 
held by others, and especially by the class teachers in 
my own school, for at almost every conference with 
my staff this question arises strongly. 

"It seems impossible to train children by much 
individual work in class by inductive methods, much 
questioning and the consequent necessary waiting 
for the child's expression to be formulated in a suffi- 
ciently acceptable form, and at the same time to get 
through the course set in a given time, and especially 
properly to prepare the children also for examina- 
tion purposes. 

' ' For instance, in an illustration given of obtaining 
from the class the definition of the diameter of a 
circle, the time taken, if similarly applied to other 
parts of the course, would not permit of a present 
average syllabus being more than about half com- 
pleted, nor would the information got be available 



INTRODUCTION. 15 

throughout the class for reproduction at a more or 
less distant examination. 

"Another illustration: Two years ago an inspector 
{fons et origo malorum W. H. W.), examining 
Standard V (approximately American Grade VI), 
asked for a definition of a proper noun, and, not get- 
ting a satisfactory answer, tried to obtain it from the 
boys with the aid of many questions and illustrations. 
He took up twenty minutes of the lesson, and failed 
in the end to get what was wanted. 

"Of course, all the time the children were being 
educated on the best lines; they showed eagerness, 
interest and active thought. (This, I fear, is a con- 
cession to the lecturer as a theoretical person. W. 
H. W.) But, the time taken, in view of the rest of 
the syllabus, was excessive, and the result at the end 
was not satisfactory." 

Then follows a paragraph in which the writer ex- 
presses his willingness to carry out the new methods 
provided the educational authority will dispense 
with tangible results. 

This is a strong letter. It expresses views which 
are very common, and which, moreover, are held by 
some of the most successful schoolmasters I know. 
And they will have to be met by educational science. 
I, myself, believe that, until these questions are dealt 
with in such a way as to be acceptable to teachers, the 
breach between theory and practice will remain. 
Professors, teachers of method and insx^ectors may 
continue as now to receive lip-homage for the meth- 
ods they advocate, but their directions will be hon- 
ored rather in the breach than in the observance. In 
actual practice there will be little, if any, change. 



16 INDUCTIVE VS. DEDUCTIVE METHODS. 

What, then, do I suggest I I propose the experi- 
mental determination of these disputed questions in 
the schools themselves. There is an increasing num- 
ber of teachers willing — nay, anxious — to carry out 
such experiments if adequate guidance be given to 
them. To the description of one attempt at an expe- 
rimental solution of some of these disputed points I. 
now proceed. 



II. THE PROBLEM OF THE EXPERIMENT. 

No one can liope to solve all the questions raised 
in the never-ending controversy about inductive and 
deductive methods by means of a single experiment 
or by means of a, single series of experiments. Yet, 
if one attempts to deal with the matter experimen- 
tally, one must deal with some definite subject-mat- 
ter. There is danger in this, since we may find out 
afterwards that our conclusions are true only for 
subject-matter of that particular kind; but it is a 
danger which must be faced. 

A good plan, if one is conscious of bias, is to select 
subject-matter which favors the chances of the 
method in which personally one does not believe. So 
I, believing in inductive rather than in deductive 
methods, chose geometrical definition as the subject- 
matter for my experiment. 

There is much good opinion in favor of deductive 
treatment of definitions, especially when, as in 
demonstrative geometry, a sort of system of 
reasoned conclusions is to be built up upon those 
definitions. It is argued that a jDupil ought to know 
exactly what the definition means, that the exact 
wording of it is very important for that purpose, 
and that at some stage in the procedure the defin- 
itions should be memorized. There is no question 
here about the introduction to demonstrative geom- 
etry. It is supposed by both parties to the dispute 

17 



18 INDUCTIVE VS. DEDUCTIVE METHODS. 

that manual work and geometrical construction are 
necessary propaedeutics to any rational system of 
geometry. But if we are ever to have a system of 
demonstrative geometry we must, it is said, have 
exact meanings for our terms or we shall never be 
able to reason in words at all. 

This is by no means a weak case, and to it addi- 
tional importance is given at the present juncture, 
when so much dissatisfaction is being expressed as 
to the 'chaos' into which geometrical teaching is 
falling through what is alleged to be an excess of 
inductive method. 

Those who advocate purely inductive methods 
urge that the memorizing of the definition and the 
study of its application to examples is not, in the 
truest sense, knowing the definition ; it is urged that 
it can be known better if it is built up from the ex- 
amples. It is asserted that the memorizing of defini- 
tions leads to bad errors, 'howlers,' of which the 
following, once given to me, is a choice example : "A 
circle is a figure bounded by a straight line, which is 
such that every point within it is at equal distance 
from every other point." There is a tendency to 
concede that inductive methods are slow; there is 
some tendency also to concede the point that induc- 
tive methods will not prepare a pupil so well for 
examination purposes. But it is argued that what 
he does learn he will remember longer, and that he 
will be made more intelligent. 

The use of the word 'intelligent' in educational 
disputes amounts almost to a public scandal, and I 
do not propose to use it without giving an opponent 
some way of checking the assertion. 

By intelligence, in this case, I am going to mean 



THE PROBLEM OF THE EXPERIMENT. 19 

the power gained to deal with new material in con- 
sequence of the mental processes which the pupil has 
passed through in acquiring the old. I have found 
this interpretation of the word acceptable to both 
parties in the dispute. 

Let me now endeavor to disentangle the threads 
and see how far the assertions made, first on one side 
and then on the other, are susceptible of experimen- 
tal determination. 

First of all, we can quite easily find out whether 
pupils taught inductively or deductively know the 
required definitions better immediately after they 
have acquired them. We shall demand exact mean- 
ings, but not a stereotyped form of words. A child 
taught inductively would not, of course, know a par- 
ticular form of words for a definition, but no devia- 
tion from accuracy of meaning must be allowed. 

Secondly, we can find out, by repeating the exer- 
cises later on, whether the pupil forgets more or less 
after one method than after the other. We are thus 
testing the durability of his knowledge. 

Thirdly, we can find out how many 'had errors' 
are made by pupils who are taught by inductive and 
deductive methods, respectively. 

Fourthly, we can find out which of the two methods 
gives the pupil the greater power of attacking new 
material successfully. 

In so far as we can determine these issues, we are 
in possession of the clues which will lead us to 
reasonable conclusions on most, if not all, of the 
questions raised in this section and in the section 
entitled Introduction. 



III. THE GENEEAL PLAN OF THE EXPERI- 
MENT. 

One difficulty in all work of this kind is to find 
some unsophisticated material with which to experi- 
ment. I wanted to work, if I could, with some school 
children who had never learnt or even heard of a 
geometrical definition throughout the whole of their 
school lives. I think I succeeded in getting this con- 
dition fulfilled with some Standard V girls in the 
southwest of London and some Standard III boys in 
the southeast. 

In the course of the experiment one of the boys' 
fathers told him it was Euclid, which it wasn't, and 
gave him one or two 'tips' which spoiled some of his 
papers ; but, with that exception, nothing transpired 
to indicate that we were not working on virgin soil. 
I worked also with a Standard VI and VII girls' 
class in a poor school. These girls, though not au 
courant with geometrical definition, had nevertheless 
done much constructive work and were accustomed to 
express themselves freely and exactlj^ I worked also 
with a Standard VII boys' class in a poor school. 
There was a little difficulty here with one or two of 
the definitions, owing to the boys' attendance at the 
Manual Training Center, where they had learnt 
something about them. And, finally, the work was 
done with an ex- VII boys' class, the highest class of 
a higher grade school. The teacher of this class was 

20 



GENERAL PLAN OF THE EXPERIMENT. 21 

a man who had for j^ears attended lectures on psy- 
chology, and was accustomed to teach very largely 
by inductive methods. 

I think it will be conceded that we have here a good 
variety of material; and, since in every case we 
worked with all the pupils of the class, and not 
merely with selected pupils, it will also be conceded 
that if any tendencies show themselves throughout 
the whole range of our material, the probability that 
they are chance results is very small indeed. So 
much for a general survey of the material; it re- 
mains to be seen how it was utilized. Briefly, though 
there were local variations in procedure, the same 
general plan was followed throughout. 

A whole class, under one teacher and working un- 
der the same syllabus of instruction, was divided into 
two equal groups. The children were required to 
attempt the definition of some geometrical shapes 
which were placed before them in the form of large 
drawings, and on the results of these attempts the 
class was divided. One of the two groups subse- 
quently acquired the definitions inductively. The 
other group learnt them, but the children were in- 
structed that the exact words given them in the defi- 
nition were not required as long as they gave all the 
meaning. The two groups were tested immediately, 
and after shorter and longer intervals. And, after 
some interval of time, new material of a somewhat 
analogous kind was given to the children to define 
with a view to discovering which of the two groups 
could better apply their old knowledge, as we say, 
though it is, in some aspects, rather an application of 
a method than an application of knowledge. 

Two of the classes were taken by me in the indue- 



22 INDUCTIVE VS. DEDUCTIVE METHODS. 

tive acquisition of the definitions ; in the three other 
cases they were taken by their own teachers. 

In one of the schools in which I took the exercises 
myself one of the other teachers inquired of the 
teacher of the class whether I wished the inductive 
method to succeed. ' ' I think he does, ' ' was the reply. 
''Then," came the prompt rejoinder, ''he ought to 
have let you take it ; you would have got it into them 
much better than he would." With this unsolicited 
testimonial to my handling of the method, I proceed 
to a detailed description of the five series of experi- 
ments actually carried out in the schools above cited. 



IV. FIRST SERIES OF EXPERIMENTS. 

1. General characteristics of the children who 
worked the exercises. 

This experiment was carried out during the months 
of September, October and November of the year 
1911. The work was done with the whole of a Stand- 
ard V class of girls of an average age of 11 years 8 
months. The children knew nothing about geomet- 
rical definitions and were not biased by practice or 
novelty towards either of the methods employed. It 
is, of course, necessary to know the customary lines 
of teaching in a school in order to prevent one from 
drawing misleading conclusions from the results of 
experiments of this kind. The infant school work 
which these girls had done some four, five or six 
years previously was very little affected by tend- 
encies to the kindergarten or sensory type of instruc- 
tion. . The school was situated in a neighborhood 
slightly above the average for municipal elementary 
schools, and the girls of the class were accustomed 
to give full attention to their school work. 

2. The Preliminary Tests. 

Drawings of squares, triangles, oblongs and diam- 
eters of circles were placed upon large blackboards, 
with their names written above them, thus : 

23 



24 INDUCTIVE VS. DEDUCTIVE METHODS. 




^ 



Wu,ayTj&a 





qV -r-C<Vr> ^-^^fl/ 





<t^ 



0-^. 



T 




.^U^i^r^e'CM^ f>f G.'i^C^C^ 






FIRST SERIES OF EXPERIMENTS. 25 

The children had the drawings pointed out to 
them, with the accompanying words, ''These are 
squares," etc. In the case of the hist set of figures 
tlie straight line was pointed to with the words, 
"This is a diameter of a circle, and this is a diameter 
of a circle," etc. Then upon a blackboard the fol- 
lowing questions were written : 

1. What is a square? 

2. What is a triangle? 

3. What is an oblong? 

4. What is a diameter of a circle? 

All the children in the class were required to an- 
swer the questions. They were told to do so thought- 
fully and without hurry. No time limit was imposed 
in this or in any other of the subsequent exercises of 
this experiment. 

3. The Method of Marking the Preliminary Tests. 

The importance of this question merits a short dis- 
cussion of the principles involved. I suppose one's 
first notion is something like this: Let us just take 
any currently accepted definition of a square, etc., 
Euclidean or other, and mark the children's exer- 
cises in accordance with their correspondence or non- 
correspondence with that. It will not matter very 
much which definition we take, provided that we keep 
to the same one all through the marking. 

A perusal of the answers shows us immediately, if 
we had not known it before, that this method will 
not do. The children will hardly be likely, for ex- 
ample, to write down that a square is a rectilinear 
quadrilateral with a right angle, as one good defini- 
tion gives it. They know the meaning of a straight 



26 INDUCTIVE VS. DEDUCTIVE METHODS. 

line ; they know the meaning of four sides ; they do 
not know the meaning of right angle, and if they did 
they would tell you quite candidly, if they had been 
properly taught, that the definition just given was 
wrong. A square has four right angles, not oyie, they 
would urge. The teacher might quite authoritatively 
inform them that every four-equal-sided straight- 
lined figure, if it have one right angle, must have 
four ; therefore, why say four in the definition % and 
that a definition should not say more than it need. 
I hope the teacher won't, because the redundancy 
is no redundancy to the child at this stage ; indeed, is 
no redundancy at all until the child is in a mental 
condition to deduce some of the properties from the 
others. Then, and only then, can one strike out the 
derivable properties and be satisfied with the others 
as sufficiently defining the term. This is an exceed- 
ingly interesting exercise, but its time is not yet. 
''Well," I can hear an impatient mathematician say, 
"this isn't mathematics; this is psychology'; the chil- 
dren are not going to be marked on real definitions 
at all." On the contrarj^ I assert that they are going 
to be marked on the very realest of definitions ; they 
will be marked according to the number of qualities 
and properties which they can themselves see to be 
common to all the specimens called by the same name. 
And I go further, and assert that the mathema- 
tician's definition, suitable and right for those who 
know the system of knowledge within which the defi- 
nition finds a place, is just mere arbitrariness to 
those who do not. And, moreover, to tell the child 
that he may mention some of the coimnon properties 
that he finds, and that he may not mention others, un- 
less he is in a position to see for himself that some 



FIEST SERIES OF EXPERIMENTS. Zl 

are derivatives or derivable, is to shut him up sliarp 
before a mystery. He won't do much spontaneous 
defining after that. Perhaps, some day, we may have 
a system of demonstrative geometry built up by psy- 
chological research. If so, the definitions will change 
as knowledge accumulates and reasoning becomes 
more penetrating, just as our definitions do of the 
common objects of daily life. Imagine a system of 
geometry for school cliildren with evolving defin- 
itions. It is doubtless too horrible, and I do not, at 
present, ask my reader to accept such a thing, but 
only to grant that if I want to mark fairly the efforts 
of untaught children in spontaneous definition I must 
be guided by what they do, and not mark on an a 
priori scheme, settled beforehand by Euclid or by 
another geometer, or by myself, who am no geometer. 

What, then, do the children say in answer to these 
questions, "What is a square?" etc. There are be- 
fore us between forty and fifty sets of answers, and, 
though it would be illuminating for anyone to read 
the whole of them, they cannot be reproduced here. 
The interest in them lies in the fact that they repre- 
sent untaught, spontaneous attempts; it is an inter- 
est which is at first psychological; the pedagogical 
interest comes later. Perhaps one of the best papers 
and one of the poorer ones may be found worthy of 
attention. I request the reader to look at the draw- 
ings whilst considering the children's definitions. 
The specimen papers follow exactly as they were 
written : 

Nellie W., aged 12 years 3 months, wrote : 

1. A square is an object with four corners and four lines, two 
for the sides and one for the top and bottom. 

2. A triangle is an object with three corners, and three lines, 
two slanted ones and one straight one at the bottom. 



28 INDUCTIVE VS. DEDUCTIVE METHODS. 

3. An oblong is an object with four corners and four lines, the 
lines at the top and bottom being longer than those of the sides. 

4. A diameter of a circle is a line drawn right across the circle 
from one side to the other each line is called a diameter. 

Winnie T., aged 12 years 8 months, wrote : 

1. A square is a kind of box with four lines all the same length. 

2. A triangle is a thing with three sides not all the same length. 

3. An oblong is a thing with two short sides and two long sides. 

4. A diameter is a circle with a number of lines going from one 
side to the other. 

Even from these two papers one may get a hint as 
to the way the children are going to sum the figures 
up, and a careful search through all the papers re- 
veals that by one or another the following points of 
accurate definition are mentioned : 

Common qualities mentioned in children's defini- 
tions. 

1. A square is a shape, figure, drawing. 
It has lines or sides. 

It has four lines or sides. 

It has equal lines or sides. 

It has straight lines or sides. 

It has corners. 

It has four corners. 

It has equal corners. 

(A total of eight points.) 

2. A triangle is a shape, figure, drawing. 
It has lines or sides. 

It has three lines or sides. 

It has corners. 

It has three corners. 

(A total of five points.) 



FIRST SERIES OF EXPERIMENTS. 29 

3. An oblong is a shape, figure, drawing. 
It has lines or sides. 

It has four lines or sides. 
It has straight lines or sides. 
It has two long sides. 
It has two short sides. 
The two long sides are the same length. 
The two short sides are the same length. 
The two long sides are opposite each other. 
The two short sides are opposite each other. 
It has corners. 
It has four corners. 
The corners are all the same size.) 
(A total of thirteen points.) 

4. A diameter of a circle is a line. 
It is a straight line. 

It goes through the middle or center of a circle. 
It goes from one side or edge of the circle to 
the other. 

(A total of four points.) 

The papers were marked thus : One mark was al- 
lowed for each common attribute correctly given. It 
was decided not to allow thing or object as equivalent 
to shape, diagram, etc., for it was thought that 'thing' 
was so wide a term as to be hardly available for the 
purpose of these definitions, and that by the word 
'object' children usually meant a material thing in 
three dimensions of space. 

It is not contended that all the above units of 
marking are exactly equal in value ; it is contended 
only that marking on these units is easy, practically 
serviceable, and yields results from which one can 
draw reliable conclusions for practical purposes. 



30 INDUCTIVE VS. DEDUCTIVE METHODS. 

On the results of this marking all the children of 
the class were divided into two equal groups, one of 
the best children being placed at the head of the first 
group — Group A — an equivalent child being placed 
at the head of the second group — Group B — then the 
children next in order were placed, one in each group, 
and so on down the list, carefully preserving the bal- 
ance all the way down, till all the children were di- 
vided between the two groups. 

There is a weakness here which needs attention. 
It is not usually satisfactory to divide a class on the 
basis of one test only. It is probably much more sat- 
isfactory where the higher mental functions are con- 
cerned than it is where simple sensory functions are 
measured, but it is risky even in the former case. It 
adds enormously to the probability of a valid result 
if several tests of the same kind are taken and the 
results of these correlated. One then feels confi- 
dence, if the results of the tests correlate highly with 
one another, that one is testing some function or 
group of functions which is operating steadily, and 
that each child is working at about its 'true form' as 
compared with the others. In some of the subse- 
quent experiments of this kind I adopted that plan, 
but as I wished to be present during the whole time, 
and on each occasion when exercises were done in 
this case, I reduced the number of Preliminary Tests 
to one, but I did so with a full consciousness that I 
should feel less reliability in the equality of my two 
groups. 

The marks obtained in the Preliminary Tests by 
the two groups, respectively, will be shown in the 
section headed 'Results.' 



FIRST SERIES OF EXPERIMENTS. 31 

4. The Teaching of the Tivo Groups. 

About a week later the two groups were separately 
instructed in the subject-matter of the definitions. 
As each child had already made some attempts for 
herself under test conditions to' define the terms, she 
was in a favorable condition for the reception of 
knowledge by any method. 

I had decided that one of the two groups should 
have the definitions written out for them, with illus- 
trative drawings underneath, and that the children 
of this group should be instructed to study and learn 
the definitions. They were told that they were going 
to be examined afterwards, that they might then 
write down the exact words if they liked, but that as 
long as they got down all the meaning they would 
lose no marks because they had failed to remember 
the exact words. This resembled the way in wiiich I 
remember myself to have learnt the Euclidean defi- 
nitions. I studied the words and let my attention 
pass from words to figures to see the illustrations of 
the general statements in the definitions — it may 
fairly be called a deductive method. There is, how- 
ever, one important difference. The definitions given 
to the children are such as they themselves are capa- 
ble of making up. That does not mean that every 
child, nor even any child, in the class could say pre- 
cisely all these things by itself; it does mean that 
they are on the lines of the child's own evolutionary 
track. 

The Definitions as Learnt Deductively. 

They were written down with illustrative examples 
drawn underneath the definitions. The drawings 
were the same as those given before in the Prelimi- 



dZ INDUCTIVE VS. DEDUCTIVE METHODS. 

nary Tests ; they will not be reproduced here. The 
definitions were worded thus : 

1. A square is a shape with four sides and four 
corners. The sides are straight and they are all of 
the same length ; the corners are all of the same size. 

2. A triangle is a shape with three sides and three 
corners.* 

3. An oblong is a shape with four straight sides 
and four corners. Two sides are long and two are 
short. The two long ones are opposite to each other 
and are of the same length, and the two short ones 
are opposite to each other and are of the same length. 
All the corners are of the same size. 

4. A diameter of a circle is a straight line going 
through the middle of a circle from one side of the 
circle to the other. 

The Definitions as Taught Inductively. 

The teacher, myself in this case, having the points 
of each definition in mind, taught up to them, but no 
instruction was given by the teacher otherwise than 
by questioning. 

In another school, in which also I did the necessary 
teaching myself, a discussion arose afterwards be- 
tween the two groups of girls because one group had 
done rather better than the other in the subsequent 
testing. ''You ought to do best," said a girl of the 
deductive group to the inductive group ; ' ' the gentle- 
man taught you the definitions ; we had to learn 'em 
for ourselves. ' ' 

''That's just where you're wrong," she was an- 



*It will be remembered that one of the triangles drawn had 
curvilinear sides. 



FIRST SERIES OF EXPERIMENTS. 66 

swered; ''the gentleman never told us a thing; we 
told him all about it. ' ' 

But perhaps a little more explicitness may be use- 
ful to those who may wish to repeat the experiment. 
Let me illustrate by means of the last example : 

Pointing to all the diameters drawn, the teacher 
says: ''What can we say about all these?" The an- 
swer 'lines' will be received. He can then ask the 
question: "What is a diameter of a circle!" He 
will be answered, if he chooses his questionee well : 
"A diameter of a circle is a line." One feature of 
the method is that the teacher accepts all the answers 
given to him and translates them into concrete form. 
He draws a curved line on the blackboard, but not 
within one of the circles, and asks: "Is that a diam- 
eter of a circle?" He is answered: "No, because it 
is not a straight line. ' ' He draws a straight line, still 
outside the circle, and asks : "Is that right ? ' ' The 
answer comes: "No, because it's not in a circle." 
"Very well," the teacher says, "let us try again. 
What is a diameter of a circle f'i If he chooses a 
child's answer, as he should, from among the least 
proficient of the class, he will be answered: "A di- 
ameter of a circle is a straight line inside a circle. ' ' 
He accepts the answer and draws a straight line in 
a circle which neither passes through the center nor 
touches the circumference on either extremity. He 
again asks : " Is that right ? " He will be told : ' ' No, 
because the line does not go to the edge on both 
sides." He corrects his drawing, producing the line 
each way to the circumference. He will now be told 
his line is wrong, because it must pass through the 
middle or center of the circle. He then draws a fresh 
line passing through the center, but cutting the cir- 



64: INDUCTIVE VS. DEDUCTIVE METHODS. 

cumference, and lie is now told the Hue must reach 
the edges, but not pass over them. At this stage he 
can rely upon receiving a correct definition from the 
great majority of his pupils ; but it is essential, if we 
are to keep this method distinct from the other one, 
that he does not ask a number of children to give the 
correct definition. One or two may do so, and the 
teacher then passes on; otherwise the mnemonic 
repetitive factor comes in here as in the other 
method, and for the purposes of this experiment it is 
usually desirable to avoid this. 

I do not propose to go in detail over the method 
employed for teaching the remaining definitions. 
Any experienced teacher — and this paper is not writ- 
ten for other than experienced teachers — will be able 
to ask analogous questions and get the answers cor- 
rected in analogous ways. With children taught fre- 
quently on this method it is quite possible to get the 
necessary drawings and corrections, or most of them, 
done by members of the class, so that the machinery 
of correction and amplification is mainly in the hands 
of the class, with the teacher there to see fair play, 
and direct the discussion to profitable issues. But I 
do not press this latter point ; the work of concrete 
exemplification of error was, in the case of all the 
experiments about to be described, solely in the 
hands of the teacher. Teachers who would not kgree 
with the method of mutual correction may quite well 
agree with this. 

There are one or two points of detail, however, 
which may cause difficulty. It is of little use for the 
teacher to point to the squares drawn and ask,'' What 
are all these!" for he will naturally be answered, 
"Squares." Indeed, the word squares is written 



FIRST SEEIES OP EXPERIMENTS. 35 

above the drawings. But if he points to the squares, 
and the triangles, and oblongs as well, and asks the 
same question, he will get answers like ''drawings," 
"shapes," "figures," "diagrams." He can then 
start his detailed questioning to bring out the defini- 
tion of a square. Among his answers will very likely 
be, "A square is a shape with four lines and four cor- 
ners." It is obvious that many figures which are not 
squares can be drawn to comply with this definition, 
and the correction will proceed as before. The size 
of a corner is a difficulty for young children; they 
confuse corner with edge. It helps to ask, (pointing 
to angles of different sizes) "If these were the cor- 
ners of a room, how much sand or how many blocks 
could I put in that corner, and in that one ? " In some 
such way the size of the comers becomes thinkable to 
the young child. The question has been raised 
whether items of considerable difficulty, like this one, 
should not carry more than one mark. The theoret- 
ical justification is conceded, but it is argued that in 
practice a mark for each unit gives sufficiently steady 
and reliable results. 

Children will often give a quality which is true of 
only one or two of the squares or triangles. It is only 
necessary to point to the other ones in these cases. 

I need, perhaps, hardly say that these children, like 
the others, knew they were going to be examined im- 
mediately afterwards on the work they were then 
doing. 

5. The Immediate Testing of the Two Groups, 

As soon as the teaching of the Inductive Group was 
completed, the group which had been learning the 
definitions in another room also stopped their work ; 



36 INDUCTIVE VS. DEDUCTIVE METHODS. 

and in a third room, so tliat the environment of both 
the groups should be changed, both sets of pupils 
answered the following questions : 

1. "What is a square!" 

2. "What is a triangle?" 

3. "What is an oblong?" 

4. "What is a diameter of a circle?" 

6. The Marking of the Tests. 

The papers were marked exactly as in the case of 
the Preliminary Tests, so far as the positive units 
were concerned, but a new feature was added to the 
marking. It is well known that bad teaching and bad 
learning produce errors, and errors of a noxious kind. 
But some statements that we sometimes call errors 
in the work of school children are rather irrelevances 
and redundancies than errors. For instance, when a 
child, in defining a square, after mentioning the prop- 
erties of a square quite correctly, says: "and the 
corners are opposite each other," the statement is 
worth no positive mark, but neither is it worth any 
negative mark. Or when a child, in defining a tri- 
angle, says : ' * Some of the lines are curved and some 
are straight," though this, strictly speaking, is no 
part of the definition (which includes only the quali- 
ties common to all the triangles given), yet it can 
hardly be called a bad error. But it is a bad error 
for a child to say: "A triangle is a shape with three 
equal lines and three corners." Such an answer re- 
ceives five positive marks — one each for shape, lines, 
three (lines), corners, three (corners). But 'equal' 
receives a negative mark as a *bad error.' Again, 
when a child, in defining a square, says, amidst much 



FIRST SERIES OF EXPERIMENTS. 37 

wliicli is correct: ''The corners all come under eacli 
other," the statement is marked as a bad error. Or 
when a child, speaking of a diameter, says : "It must 
stand upright," the statement is regarded as a bad 
error. The first diameter in the drawings was up- 
right, hence the error. The definition was being- 
elaborated from a memory of the one drawing with- 
out comparison with the memories of the others. 

In the case of the experiment in this school, there- 
fore, besides giving the positive marks obtained by 
each group, I shall also give the negative marks, and, 
in addition, the positive marks with the negative 
marks subtracted from them. 

It is interesting, before turning to the section show- 
ing the results, to try to guess from our general 
knowledge of children's minds of the given ages 
(roughly from eleven to twelve), and our opinions 
as to the methods of teaching and learning, which of 
the two groups gained the more positive marks and 
which group made the more bad errors. 

7. Subsequent Testing of the Two Groups on the 
Same Subject-Matter. 

In discussions among teachers the question is fre- 
quently raised as to the relation between the quick- 
ness and the permanence of knowledge. Teachers 
are prone in theory to back the tortoise rather than 
the hare, though in practice they repeatedly prod 
the tortoise up. How far does the present experi- 
ment throw light on the matter ? Are we justified in 
supposing that a test given to two groups of cliildren, 
immediately after certain knowledge has been ac- 
quired, supplies us with comparative results which 



38 INDUCTIVE VS. DEDUCTIVE METHODS. 

will be true, say, a week later, a month later, and 
so on? 

To test tills point tlie exercise above described was 
repeated a week later. The children were not aware 
that they would ever have to do this work again. 

Then, once again, more than a month after the first 
test (the exact chronology of the experiment will be 
given later on), the test was repeated a second time. 
The papers were marked in both these cases exactly 
as in the first test, positive marks being given for the 
points remembered and negative marks for the bad 
errors. The results will enable us to see how far the 
comparisons between the groups based upon the im- 
mediate results are corroborated when the results 
for deferred reproduction are taken into considera- 
tion. Again, it is worth while to try to think the 
answer out on general principles before turning to 
the actual results. 

8. The Testing of the Two Groups on New Material. 

It will be remembered that the children of the class 
were divided on the results of a test in which they 
were required to find definitions for themselves of a 
square, etc. How far has the teaching or learning 
by different methods affected their power to attack 
new material of an analogous kind? This is one of 
the most important questions that can be asked of 
any method of teaching or learning. 

In ordinary pedagogical discussions it would be 
implied by assertions that children would be made 
more intelligent by one method than by another. To 
investigate this point experimentally the following 
tests were made : 

Drawings were shown of rhombuses, etc., and their 
names written above them, thus : 



FIRST SERIES OF EXPERIMENTS. 



39 



yf'fiff'wy^^-u.<J^, 




f^y-roLfvtyri^i 



*<.-rrv<l/ . 





\/X/f\..<ryrxy{h-tfX^!L<J 











In the drawings actually used the diagonals were continuous 
lines drawn in red. 



40 INDUCTIVE VS. DEDUCTIVE METHODS. 

Then the following questions were written on the 
blackboard : 

1. ''What is a rhombus?" 

2. ' ' What is a trapeziumi ' ' 

3. ' ' What is a rhomboid ? ' ' 

4. ''What is a diagonal of a square?" 

and the children were required to answer them in 
writing. 

9. The Marking of the New Material. 

As in the case of the Preliminary Tests, we must 
look for the basis of our marking in the papers them- 
selves, and not in any a priori scheme of values. It 
will, I think, be profitable if one or two samples of 
the children's actual answers be given before con- 
sidering the units of marking which were adopted. 

Laura B , aged 13 years 11 months, who 

worked in the deductive group, wrote : 

1. A rhombus is a shape with four sides and four corners, the 
four sides slant the same way, and the corners are the same size. 

2. A trapezium is a shape with four sides and four corners. 
The four sides are not the same length and do not slope the same 
way. The corners are not the same size. 

3. A rhomboid is a shape with foiu* sides and four corners. It 
has two long sides and two short sides, the long ones are opposite 
each other, and the short ones are opposite each other, the corners 
are opposite each other and slant the same way, the corners are 
all the same size. 

4. A diagonal of a square is a shape with four straight lines 
and four corners, the lines are all the same length and the corners 
are all the same size with a line going through the square from one 
corner to the opposite corner. 

Rhoda T , aged 12 years 11 months, who 

worked with the inductive group, wrote : 

1. A rhombus is a shape. It is made up of four sides. The 
lines are straight, but are drawn slantling, and there are four cor- 
ners joined exactly to one another. 

2. A trapezium is a shape. It is made up of four sides and are 



FIRST SERIES OF EXPERIMENTS. 41 

not the same length. The lines are straight but are drawn slaut- 
ling. There are four corners, they join together exactly. 

3. A rhomboid is a shape. It is made up of foiu* sides two short 
sides being opposite, and the two long sides the same. The lines 
are straight but are drawn slantling. There are four corners they 
join exactly to one another. 

4. A diagonal of a square is not a shape. It is a straight line, 
but drawn slantling. It- joins two corners exactly opposite one 
another, and the line must not reach over the two corners. 

Even from these two papers alone it would not be 
very difficult to make out a scheme of marking, and 
when taken in conjunction with the others, some 
forty or fifty in number, the following items of accu- 
rate description emerge: 

1. A rhombus is a shape or figure, etc. 
It has sides or lines. 
It has four (sides). 
It has straight (sides). 
It has equal (sides). 

Two sides slant the same way (are parallel), 
The other two sides slant the same way. 
Two that slant the same way are opposite each 

other. 
The other two that slant the same way are op- 
posite to each other. 
It has corners. 
There are four (corners). 
There are two big (corners). 
And there are two little (corners). 
The two big corners are opposite each other. 
The two little corners are opposite each other. 
The two big ones are equal. 
And the two little ones are equal. 
(A total of 17 points.) 



42 INDUCTIVE VS. DEDUCTIVE METHODS. 

2. A trapezium is a shape or figure, etc. 
It lias sides or lines. 

It has four (sides). 
It has straight (sides). 
The sides are unequal. 
It has corners. 
There are four (corners). 
The corners are unequal. 
(A total of 8 points.) 

3. A rhomboid is a figure or diagram or shape. 
It has sides or lines. 

There are four sides. 

The sides are straight. 

There are two long sides. 

And there are two short sides. 

The two long sides are equal. 

And the two short sides are equal. 

The two long sides are opposite each other. 

And the two short sides are opposite each 

other. 
The two long sides slant the same way. 
And the two short sides slant the same way. 
It has corners. 
There are four (corners). 
There are two big (corners). 
And there are two little (corners). 
The two big corners are equal. 
The two little corners are equal. 
The two big corners are opposite to each other. 
The two little corners are opposite to eacli 

other. 

(A total of 20 points.) 



FIRST SERIES OF EXPERIMENTS. 43 

4. A diagonal of a square is a line. 
It is a straight line. 

It is drawn from one corner to another. 
The corner to which it is drawn is opposite the 
other. 

(A total of 4 points.) 

If with this scheme of marldng in view we turn to 
Laura B 's paper, we shall see that she will re- 
ceive 5 positive marks for her definition of a rhom- 
bus; but she has two 'bad errors.' The four sides 
of the rhombus do not slant the same way, and the 
corners are not the same size. Her definition of a 
trapezium receives 7 positive marks; there are no 
negative marks for bad errors in this definition. 

It will now, doubtless, be quite easy for anyone 
with the aid of the table to assess the rest of the posi- 
tive marks. But I might, perhaps, call attention to 
the fact that there are two more 'bad errors' in this 
paper. The corners of a rhomboid are not all the 
same size, and a diagonal of a square is not a shape. 

Rhoda T 's paper contains some errors in spell- 
ing, which, of course, are not counted in experiments 
of this kind. It can be quite easily marked on the 
system given above, and I think any teacher who 
marks it will agree that there are no 'bad errors.' 

It must not be thought that children taught 
inductively make no bad errors when they apply 
their knowledge to new material. A comparison, 
however, between the number of bad errors in- 
volved in the use of the two methods will be found 
very useful later on. All the papers in all the tests 
and exercises in this school I marked myself, and the 



44 INDUCTIVE VS. DEDUCTIVE METHODS. 

marks were subsequently checked by the head mis- 
tress of the school. 

10. Chronology of the Experiment. 

All the tests and exercises, with the exception of 
the Preliminary Test, were taken on Tuesday morn- 
ings at 10.10 A. M. All instructions and teaching 
were given by myself, and I was present with the 
children during the whole time that each exercise was 
done. 

The Preliminary Test for the division into two 
equal groups was worked on September 25th, 1911. 
The teaching and learning of the first set of defini- 
tions (which occupied 17 minutes for each group) 
and an immediate test in reproduction were done on 
October 3d. The second test in reproduction was 
given on October 10th. The test to see how far the 
children could apply their knowledge or method to 
new material was given on October 17th, and the last 
test on the first set of definitions — a further test in 
deferred reproduction — was given on November 
7th. The lessons preceding the exercises were, in 
all cases, the same. For writing out what they knew 
no time limit was insisted on : the children were all 
allowed, nay encouraged, to put down as much as 
they could. A note was kept of the time taken on 
each occasion. In the test taken immediately after 
teaching and learning 25 minutes was the limit, and 
it was noticed that the Deductive Group, i. e., those 
who had learnt the definitions, were much the 
quicker. The Inductive Group were, of course, to a 
large extent, making the definitions up as they went 
along. In the first test of deferred reproduction. 



FIKST SEKIES OF EXPERIMENTS. 45 

which took place a week later, 30 minutes were taken ; 
in the test to show the power of application to new 
material, 32 minutes; and in the second test of de- 
ferred reproduction, which took place a month after 
the acquisition of the original definitions, 32 minutes 
were taken. It is possible to ascribe this lengthening 
of the time of the exercise to an increasing difficulty 
of remembrance. This may be a factor, but I am in- 
clined to think there may be another; the children 
may be getting more thoughtful over the work, and 
consequently slower. 

11. Results, 
(a) The Marks for the Preliminary Tests. 

Every mark which every child obtained in every 
exercise was carefully tabulated, though from the 
final table it was necessary to exclude three or four 
children who had been absent on several occasions. 
The first and last child in the Inductive Group were 
among these cases, so the corresponding cMldren, 
namely, the first and last of the Deductive Group, 
were also omitted. There were then 21 cases left in 
each group. 

In the Preliminary Test, in which the children 
tried by themselves to see what they could do in de- 
fining square, triangle, etc., the group which subse- 
quently did the inductive work gained an average 
mark of 9.4, with a mean variation of 2.2; and the 
group which subsequently learnt the definitions de- 
ductively obtained an average mark of 9.5, with a 
mean variation of 2.3. The highest mark in each 
group was 15 ; the lowest mark was 6. In so far as 
it is possible to make a satisfactory division of a 
class on one test only the groups were well balanced. 



46 INDUCTIVE VS. DEDUCTIVE METHODS. 

(h) The Marks for the Test Immediately After the 
Definitions Had Been Taught and Learnt. 

The Inductive Group gained an average mark of 
22.6 of positive marks, with a mean variation of 2.5, 
whilst the Deductive Group gained an average mark 
of 25.6, with a mean variation of 2.2.* This is a 
clear and significant difference in favor of the De- 
ductive Group — a difference which is accentuated 
when 'bad errors' are taken into account, for the 
former group makes 21 bad errors and the latter 
group only 12. Deducting the negative marks — the 
marks for bad errors — from the positive marks, it is 
found that the Inductive Group scores an average of 
21.6 marks (mean variation 2.6), and the Deductive 
Group scores an average of 25.0 (mean variation 
2.6) — a still clearer and more significant difference. 
It is highly probable from the above average marks 
and variabilities that this difference is a difference, 
so to speak, all along the line, i. e., one which will be 
found between both the best pupils of each group 
and also between the worst. That relationship, how- 
ever, will be shown more clearly by the following 
table : 

Table I, shoiving the work of the Inductive and Deductive Groups 
compared, in the Preliminary Test, and in the First Test after 
the definitions had heen learnt and taught {positive ma?'ks 
only). 

Group A (Inductive). Group B (Deductive). 
Av, marli 
Maries in No. in A v. mark No. Av. marlc Av. marlj 

preliminary of prelim, in first of in prelim, in first 

tests. girls. test, final test, girls. test, final test. 

12 and over 5 13.0 25.4 5 13.4 25.4 

9, 10, 11 7 10.0 22.0 7 10.0 2.5.1 

7, 8 5 7.8 21.4 5 7.6 26.6 

6 4 6.0 21.7 4 6.0 25.5 

*The difference between the averages is about five times the 
probable error of their difference, even ou the assumption that the 
series are not positively correlated. 



FIRST SERIES OF EXPERIMENTS. 47 

An inspection of Table I shows that, while there 
is no difference in the results between the two sec- 
tions of the ablest children, those at the top of each 
group, there are considerable differences in the re- 
maining sections. The argument of the teacher 
whose letter I have given, namely, that routine meth- 
ods are better for the immediate reproduction of 
the actual material learnt, must be conceded for chil- 
dren of this level of ability. A teacher once asked 
me rather scornfully: ''What did you expect from 
your 17 minutes' teaching?" Not much, perhaps, 
but we shall see more about that later. And, of 
course, there were also 17 minutes' learning, so the 
comparison was a fair one in any case. 

(c) The Marks for Tests of Deferred Reproduction. 

It is, however, one thing to answer questions im- 
mediately after one has learnt, or has been taught, 
the answers; perhaps it is quite another thing to 
give those answers accurately by and by. 

The second Final Test took place one week after 
the first Final Test, and the children, as I have 
already pointed out, did not know they were ever to 
have the exercises again. It is a matter of great 
importance to the teacher, and it is a matter of 
great importance to the experimenter to know how 
far the immediate results from the work of any 
group of children may be taken as fairly represent- 
ing what that group of children will do later on. 
And, perhaps, a week is too short a time. ''The 
children won't have forgotten all about it by then," 
as a teacher said, so a third Final Test — the second 
test of deferred reproduction — was given a month 



48 INDUCTIVE VS. DEDUCTIVE METHODS. 

later. Let me present a few comparisons. I will 
give first the positive marks only : 



Taltle II, shoxclng the relation hetiveen 


immediate and 


deferred 


reproduction (positive 


marks oi\ 


ly). 




Prelimi- 




First 


Second 


Third 


nary 




fiual 


final 


final 


test. 




test. 


test. 


test. 


Inductive Group. Average.... 0.4 




22.G 


23.2 


22.0 


M. V 2.2 




2.5 


2.0 


3.1 


Deductive Group. Average 9.5 




25.6 


25.4 


24.9 


M. V 2.3 




2.2 


2.1 


1.9 



I will next sliow the immediate and deferred re- 
sults for tlie two groups, when the negative marks 
for the Final Tests have been subtracted from the 
positive marks : 

Table III, shoicinr/ the relation hctwcen immediate and deferred 





reproduction. 










Prelimi- 


First 


Second 


Third 




nary 


final 


final 


fiual 




test. 


test. 


test. 


test. 


Inductive Group. 


Average.... 9.4 


21.G 


22.2 


21.0 




M. V 2.2 


2.6 


2.2 


3.0 


Deductive Group. 


Average. ... 9.5 


25.0 


24.9 


24.1 




M. V 2.3 


2.6 


2.6 


2.1 



The results from both the tables are in marked 
agreement. Both groups of children have gone down 
somewhat, but the children taught inductively have 
lost less of their original knowledge than the group 
which worked deductively. It appears that for 
groups of children of this age and ability the imme- 
diate results of these methods of learning and teach- 
ing may be accepted as indicating, comparatively, 
not only what the children can do at the time, but also 
what the groups will do by and by. 

This may be shown roughly in the following table : 



FIRST SERIES OF EXPERIMENTS. 49 

(d) Correlation Between Immediate and Deferred 
Reproduction. 

Table IV, showing the correlation hetween immediate and defended 
reproduction in the two groups, section hy section {positive 
marks only). 

Group A (Inductive). Group B (Deductive). 

Av. Av. Av. Av. Av. Av. 

mark mark mark mark mark mark 

Marlis in No. first second third No. first second third 

preliminary of final final final of final final final 

test. girls, test. test. test, girls, test. test. test. 

12 and over 5 25.4 25.6 2.5.2 5 25.4 24.8 25.0 

9, 10, 11 7 22.0 24.0 2.3.4 7 25.1 24.4 23.6 

7,8 5 21.4 20.8 19.0 5 26.6 26.2 25.2 

6 4 21.7 21.7 19.0 4 25.5 27.0 26.2 

It is fairly obvious that positive correlation exists 
between the children's immediate work, their work 
after one week's interval, and their work after a 
month's interval. But there is some irregularity 
here and there, so that it will be better to set down 
each child's individual results and work out the cor- 
relations between them rather than trust wholly to 
the inspection of the averages of these corresponding 
sections. The lists were arranged thus : 

Inductive Oroup (positive marks only). 

Name. First final Second final test. Third final test. 

(Initials only.) test. A week later. A month later. 

E. S 29 24 26 

M. J 27 27 22 

D. S 21 . 24 23 

E. B 25 25 27 

E. vi.... I.. .......... 20 20 17 

(21 cases.) 

A similar table was made of the individual results 
from all the children who worked in the Deductive 
Group, which also contained 21 cases.* Then the 
Pearson coefficient of correlation, or 'r' formula, 



50 INDUCTIVE VS. DEDUCTIVE METHODS. 

which runs thus : r = , was applied to the in- 

IlaiO-2 ' ^^ 

dividual cases, and the following results were ob- 
tained : 

Inductive Deductive 
group. group. 

Correlation between results of first and 

second final tests + .537 + .532 

Correlation between results of first and 
tbird final tests +.589 +.331 

It certainly appears that whilst the averages for 
the groups have remained remarkably steady, the 
coefficients of correlation show that the individual 
children have changed places considerably. But we 
must, I think, admit that an immediate test of the 
result of a method of teaching or learning is one 
which gives us reasonable ground to expect, from the 
group as a. whole, a similar result later on. That is, 
for reproductive exercises, where fairly homogene- 
ous groups of school children are concerned, tests of 
immediate reproduction and tests of deferred repro- 
duction give very similar results. We must appar- 
ently concede the point argued for in the head mas- 
ter's letter previously quoted, viz., that a mechan- 
ical method is better either for immediate or de- 
ferred reproduction at examinations. Let us concede 
that point, bearing in mind that our examination 
results, so far, have been only of a kind in which the 
exact reproduction has been asked for of precisely 
what was taught or learnt. 

(e) Results When the Two Groups Are Tested on 
Neiv Material. 

We have seen, hitherto, that the Deductive Group 
has scored higher marks than the Inductive on every 



FIRST SEEIES OF EXPERIMENTS. 5.1 

occasion. Certainly our preliminary division of the 
two groups on one test only is open to criticism, and 
we might suppose that we had not really succeeded 
in obtaining equal groups. And possibly we have 
not, though I think it likely that they were approxi- 
mately equal in initial capacity. Whether that be 
the case or not, there is found a definite difference 
between the two groups on the results of a test on new 
material, and in the opposite direction from the pre- 
vious difference. The group taught inductively now 
leads the way. Counting only positive marks, the 
difference is small: the Inductive Group scores an 
average mark of 20.9 (mean variation 3.8), whilst 
the Deductive Group scores an average mark of 20.0 
(mean variation 2.7). But when the marks for 'bad 
errors' are subtracted there is a distinct and decided 
difference: the Inductive Group scores an average 
mark of 19.4 (mean variation 3.5), whilst the Deduc- 
tive Group scores an average mark of 17.7 (mean 
variation 2,5).* The more mechanical method has 
produced a much larger crop of 'bad errors,' for 
there is an average of 2.3 per child in this group, 
against an average of 1.2 in the other : 

Tahle V, shoxcing the 'had errors' of the two groups. 

Inductive Deductive 

group. group. 

No. of girls with 5 bad errors 2 

No. of girls with 4 bad errors 3 

No. of girls with 3 bad errors 3 3 

No. of girls with 2 bad errors 5 7 

No. of girls with 1 bad error 6 4 

No. of girls with bad errors 7 2 



*The difference between the means is more than twice its prob- 
able error, even on the assumption that the series are uncorrelated. 



52 INDUCTIVE VS. DEDUCTIVE METHODS. 

For children of this age and capacity, therefore, 
we are entitled to urge that the inductive method is 
much less provocative of error when 'new material' 
is given for test purposes. On the old material the 
Inductive Group made more errors than the Deduc- 
tive in every test, so that we cannot suppose that the 
result with new material is consequent upon a 
greater initial tendency to 'howlers' in the De- 
ductive Group ; indeed, the tendency seems the other 
way, if there be one. We are often warned that 
averages are prone to conceal important differences 
between individuals, but I cannot expect huge tables 
of individual results to be printed, and, indeed, no 
useful conclusions in experiments of this kind could 
be drawn from individual results as such if they were 
printed ; they must be grouped if they are to be of 
service. But I can show how far the superiority 
holds or fails for the sections of corresponding ini- 
tial ability in the two groups : 

TahJe VI, sJioii'ing the work of the tico groups compared, section 
by section, in the last reproductive test, and, in the test of 
application to new material, the marks for had errors being 
sxihtracted from the positive marks. 

Group A (luductive). Group B (Deductive). 











Av. 






Av. mark Av. mark 


Av. mark 


mark ap- 


Marks in 


No. 


last application No. 


last 


plication 


preliminary 


of 


reprodnc- to new of 


reproduc- 


to new 


test. 


girls. 


tivetest. material, girls. 


tive test. 


material. 


12 and over. 


. 5 


23.8 20.4 5 


24.4 


17.6 


9, 10, 11 


. 7 


22.8 21.7 7 


22.9 


17.0 


7, 8 


. 5 


18.6 17.4 5 
18.2 18.2 4 


25.2 
25.7 


19.0 





. 4 


las 



In the case of every corresponding section the loss, 
when application is made to new material, is greater 
in the Deductive Group than in the Inductive Group. 



FIRST SERIES OF EXPERIMENTS. 53 

12. Pedagogical Conclusions. 

I have previously pointed out that, as a whole, the 
Inductive Group gains higher marks for application 
to new material than the Deductive. What pedagog- 
ical conclusions may we draw from this? Two out- 
standing conclusions seem to me to follow from this 
work: the first relating to teaching, the second re- 
lating to examination. 

First, as to the method in teaching : I suppose no 
teacher would desire us nowadays to favor a method 
merely because it enabled us to produce a better re- 
sult in the exact reproduction of what had been 
taught or learnt. Let us consider exact reproduction 
as so much to the good, but let us also remember that 
individual lessons form, or should form, part of a 
course, and the method which enables a pupil to make 
the best attack on new analogous material is, one 
may reasonably suppose, likely to emerge triumph- 
ant at the end of the course. Such a method does 
really train 'intelligence,' in the best sense of that 
much-abused word. 

Secondly, as to method in examination. Whenever 
examiners set work to be done which is a mere repro- 
duction of what the children have been taught or 
have learnt, they are favoring the mechanical 
method rather than the inductive one. 

I do not say there is no place for examination of 
that sort, but high assessments for teaching should 
never be given on such a basis. The supreme test of 
good teaching is the power, on the part of the pupils, 
to attack 'new material.' One word, however, of 
caution. I do not mean material wholly new, as many 
psychological tests are. By the use of such tests as 



54 INDUCTIVE VS. DEDUCTIVE METHODS. 

those we are measuring natural ability rather than 
the result of pedagogical work. The material should 
be ^neiv,' but it should be analogous to the work 
which the pupil may reasonably be expected to have 
done before. 

I venture to suggest that examinations of this kind 
would raise the tone and method of teaching rather 
than, as too often has been and is the case, tend 
to depress them. I wish to exempt junior scholar- 
ship examinations. They should, in my judgment, be 
psychological in the sense given above. 

But, perhaps, the reader may feel that I am build- 
ing up a huge structure of theory on the basis of a 
very little experiment ; so I will turn to the second 
series of experiments in this research and show how 
far the facts and conclusions resemble those of the 
first. 



V. SECOND SERIES OF EXPERIMENTS. 

1. General Plan. 

As before, a whole class of children, under the 
same teacher, studying the same curriculum in ac- 
cordance with the same time-table of school work, 
was divided into two equal groups on the basis of 
preliminary tests in geometrical definition, which the 
children attempted, untaught and unaided. Then 
the pupils of one group were taught inductively how 
to arrive at the definitions, whilst the other group 
learnt the definitions deductively. An immediate 
test was given to show which method was the better 
for the purposes of immediate reproduction, and sub- 
sequent tests were given on the same subject-matter 
to indicate which of the two groups was the more 
successful in deferred reproduction. Also, as before, 
a test was given on new analogous material to see 
which of the two methods showed the greater ' trans- 
fer' effect. One or two outstanding points of differ- 
ence between the conditions of this experiment and 
those of the first experiment may, perhaps, usefully 
be mentioned here before proceeding to the details. 
The children who did the work were, as before, girls 
belonging to an elementary school in London. But 
in this case they were of a poorer social class ; they 
were older than the girls in the previous school, their 
average age amounting to rather more than 13 years, 

55 



56 INDUCTIVE VS. DEDUCTIVE METHODS. 

and, measured by school standards, they were more 
proficient mentally, for these children were graded 
as Standard VI, a, and VII, whereas, in the previous 
school, the children were graded as Standard V. The 
children of this class had done a great deal of manual 
constructive work, and were taught hj a teacher 
from whom they had learnt to express their meaning 
in direct, simple language. Like the girls of the pre- 
vious school, they knew nothing of geometrical defi- 
nition; and 'Demonstrative Geometry,' or even 'Eu- 
clid, ' were terms of no meaning either to themselves 
or to their parents. 

There was, too, an important difference in the 
early part of the procedure of this experiment from 
that of the previous one. Instead of dividing the chil- 
dren on the basis of one test, four tests were given, 
in which the same material was employed through- 
out. It was believed that the division into equal 
groups would be much more satisfactory if it were 
effected on a wider basis than the results of one test ; 
and, indeed, the greater regularity of the subsequent 
work showed that to be the case. Thirty-seven chil- 
dren started the experiment, but unavoidable ab- 
senses from school reduced the number available for 
the tabulation in 'equal groups' to 34. 

2. The Preliminary Tests and the Method of 
Marking. 

As in the previous case, drawings of squares, tri- 
angles, oblongs, and diameters of circles, with their 
names appended, were shown to the children, and the 
questions, "What is a square?" etc., were written 
on the blackboard. The units of marking, as before. 



SECOND SERIES OF EXPERIMENTS. 57 

were obtained from a careful study of the answers 
actually given by the children. 

One or two instances of the children's spontaneous 
attempts at definition may possess psychological in- 
terest. 

Lily H , aged 13 years 6 months, a girl graded 

as Standard VI, a, wrote : 

1. A square is four lines each of the same length all joining 
one another and when they are joined they form a square with 
four angles the square may be straight up or slanting. 

2. A triangle is a three line drawing, joining each at the ends 
and when it is drawn it forms a drawing with three angles each 
of the same size. 

3. An oblong is a figure with four lines same as the square only 
there are two long lines and two of them are short lines with four 
angles of the same size. 

4. A diameter is the line drawn through a circle to separate 
one half from the other only it must be drawn through the centre. 

The above is one of the better papers which were 
worked during the Preliminary Tests, but it is cer- 
tainly not the best. Let me now give an inferior one. 

Ada B , aged 13 years 1 month, also graded as 

Standard VI, a, wrote : 

1. A square is a thing with four equal sides. A square can be 
all different shapes as long as the four sides are equal. 

2. A triangle is something which has three sides and the sides 
must be as long as each other. 

3. An oblong has four sides, two of the lines are short and two 
are long. The two long lines must face each other, and the short 
ones must be the same length as each other. 

4. A diameter of a circle is a round ring divided into half. 

It was seen, after a careful perusal of the chil- 
dren's papers, that the units of marking which had 
been worked out for use in the previous school were 
also quite suitable for this one. Turning to the an- 
swers of Lily II., and marking with these units, we 
find that in her first answer she has a mark for 



58 INDUCTIVE VS. DEDUCTIVE METHODS. 

'lines,' another for 'four' (lines), a third for 'of the 
same length,' a fourth for 'angles,' and a fifth for 
'four' (angles). The lines do not exactly all join one 
another, but the statement was not considered an 
error ; and the further statement, ' the square may be 
straight up or slanting' was considered irrelevant. 
In the second answer Lily H. receives a mark for 
'drawing,' one for 'lines,' one for 'three' (lines), one 
for 'angles,' and one for 'three' (angles) ; a total of 
five marks. She describes the angles as being all of 
the same size, which is certainly a serious error, and 
was probably due to the confinement of her attention 
to the equilateral triangle, which was one of the tri- 
angular figures shown. For her definition of oblong 
she receives a mark for 'figure,' another for 'lines,' 
a third for 'four' (lines), a fourth for 'two long 
lines,' a fifth for 'two short lines,' and a sixth and 
seventh for 'angles' and 'four' (angles). For her 
fourth definition, namely, that of a diameter of a 
circle, she receives a mark for 'line,' and another for 
'drawn through the center.' Lily H. thus receives a 
total of 19 positive marks. 

Ada B 's paper, marked in the same way, re- 
ceives a total of 11 positive marks. Her third an- 
swer — the definition of an oblong — is quite unex- 
pectedly good, considering the weakness of her defi- 
nitions of 'triangle' and 'diameter.' AVith various 
triangles, mostly scalene, exposed before her eyes, it 
was certainly a 'bad error' to say "the sides must be 
as long as each other. ' ' Probably, with these explan- 
ations, the method of marking adopted will be read- 
ily applicable. I will now set out the chronology of 
the whole experiment. 



SECOND SERIES OF EXPERIMENTS. 59 

3. Chronology of the Experiment. 

First of all, a preparatory exercise was given at 
9.40 A. M. on Friday, October 20th, 1911, under test 
conditions, to accustom the children to work of this 
kind, which was quite new to them. Then, on Tues- 
day, October 24th, Wednesday, October 25th, and 
Friday, October 27th, at 9.40 in the morning, imme- 
diately after Scripture lesson, a second, third and 
fourth Preliminary Test were given. On the results 
of the second, third and fourth tests the class was 
divided into two equal groups. Then on Friday, No- 
vember 3d, at the same time in the morning, the 
pupils of Group B were taught the definitions in- 
ductively by me in the way previously explained, 
whilst Group A learnt the definitions by studying 
them as written out, and referring to the illustrative 
figures drawn beneath the written definitions. I took 
care that the children who were learning the defini- 
tions should receive all their instructions from me, 
and informed the children of both groups that they 
would be required to answer questions immediately 
afterwards about what they were learning or being 
taught, respectively. The children who had been try- 
ing for themselves without help for four exercises to 
see what they could do in the way of spontaneous 
definition were, of course, in a state of receptivity 
for instruction of either kind, inductive or deductive. 
Their marks had risen steadily day by day, so that 
they were still in the 'improving' stage for this work. 
The teaching lasted 17 minutes, and, of course, the 
same time was allotted to the study of the written 
definitions. The two groups were put together imme- 
diately, and the old questions : "What is a square?" 



60 INDUCTIVE VS. DEDUCTIVE METHODS. 

etc., were written on the blackboard. It was noticed 
that the children working in the deductive group had 
answered their questions some three to five minutes 
sooner than the girls in the inductive group. 

Four days later, on Tuesday, November 7tli, a sec- 
ond test of precisely similar nature was given to 
both groups at 9.40 A. M., after Scripture lesson, as 
before. 

On Friday, November 10th, at the same time in the 
morning, and after the same lesson as on previous 
occasions, a test was given to both groups on new 
analogous matter to test the comparative 'transfer' 
values of the two methods of learning. 

The last test in the series was given at the same 
time in the morning, and after Scripture, as before, 
on Friday, December 1st. In this test the previous 
questions on the material actually studied were given 
again — ''What is a square?" etc. The object of this 
test — the second test in deferred reproduction — was 
to discover, if possible, which of the two groups had 
lost the more after a considerable interval ; in other 
words, which method of teaching or learning favored 
the more permanent retention. I ought to say that, 
with the exception of the test given immediately 
after the teaching and learning, the children were 
not aware that they were going to do any of these 
tests before they were actually set to do them. 

4. The Final Tests and the Method of Marking. 

Three of the Final Tests were repetitions of the 
Preliminary Tests, and the same method of marking 
was adopted in them as in the Preliminary Tests. 
These were the tests given immediately after teach- 



SECOND SERIES OF EXPERIMENTS. 61 

ing and learning and the two tests of deferred repro- 
duction. The remaining test, in which the children, 
without further teaching, were required to attack 
new material, was identical with the corresponding 
test given in the previous school, and it was marked 
in the same way. Drawings of rhombuses, trapezi- 
ums, rhomboids and diagonals of squares, with their 
names appended, were shown to the cliildren, and 
they were required to answer in writing the ques- 
tions : ' * What is a rhombus ? ' ' etc. 

5. Results of the Experiments, 
(a) Results of the Preliminary Tests. 

The marks obtained in the Preparatory Test will 
not be given ; it was noticed that though the children 
at the top and bottom of the lists remained much the 
same, a considerable number of children changed 
places from the Preparatory to the first Preliminary 
Test. As it was very important that the work of the 
children should be 'steady' before the class was di- 
vided into two equal groups, two more tests — the 
second and third Preliminary — were given and the 
results correlated. The work started with 37 chil- 
dren, but two had been absent during the tests, so 
they were excluded from the lists. 

The results of the three Preliminary Tests are 
shown compendiously in the following table : 

Table VII, showing the correlation between the results of the First, 

Second and Third Preliminary Tests. 

Marks in first No. of Av'age Marks in Preliminary Tests. 

preliminary t€st. children. First test. Second test. Third test. 

19, 18 7 18.3 18.7 19.0 

17. 16 5 16.4 18.4 17.6 

15, 14 9 14.7 16.4 17.0 

13, 12 7 12.7 13.9 14.7 

11 to 8 7 9.6 10.1 12.3 



62 INDUCTIVE VS. DEDUCTIVE METHODS. 

It is obvious from Table VII that high positive 

correlation exists between the results of the first, 

second and third Preliminary Tests, and that we are 

measuring a mental function, or group of mental 

functions, which is working very steadily. A precise 

numerical value for the coefficient of correlation has 

been worked out from the individual cases by means 

2xy 
of the Pearson formula r = ^— , and 'r' has been 

found to be + .80, with a probable error of .04, for 
Tests 1 and 2, and + .77 (probable error .05) for 
Tests 2 and 3. It seems very likely that a division 
of the class into two equal groups on the basis of 
such regular results as these will be satisfactorily 
effected. 

The children were divided into two groups of 17 
girls each, thus (N. G., the girl at the bottom of the 
list, was omitted) : 

Table VIII, showing the division into two equal groups. 

Group A, 

/^Marks for Preliminary Tests.^v 

Name. First Second Third Total 

(Initials only.) test. test. test. marks. 

W. F 18 24 21 63 

H. L 18 20 19 57 

H. G 17 20 17 54 

G. E 11 11 11 33 

T. F 8 9 13 30 

Averages 14.2 15.8 16.3 46.3 

M. V.'s 2.3 3.0 2.6 



SECOND SERIES OP EXPERIMENTS. 63 

Group B. 

r-Marks for Preliminary Tests.-> 

Name. First Second Third Total 

(Iidtials onlj'.) test. test. test. marks. 

D. A 17 22 22 61 

W. E 19 20 19 58 

n. L 18 17 20 55 

A. R 10 10 11 31 

L. L 11 7 13 31 

Averages 14.5 15.5 16.3 46.3 

M. V.'s 2.6 2.8 2.2 

Care was taken also that the children should be so 
arranged in the grouping that the ages of the one 
group should very closely approximate to those of 
the other. The average age of Group A worked out 
to 13 years 1 month (mean variation 7.2 months), 
and of Group B to 13 years 0.5 months (mean varia- 
tion 5.6 months). 



(b) Results of the Tests in Immediate and Deferred 
Reproduction. 

It now remains to be shown what marks were ob- 
tained after the one group had been taught the defi- 
nitions and the other group had learnt them. The 
total marks will be shown for the three Preliminary 
Tests, with the marks for immediate reproduction 
and for the two tests of deferred reproduction — the 
one given a few days later and the other about a 
month later than the test of immediate reproduction : 



64 



INDUCTIVE VS. DEDUCTIVE METHODS. 



Table IX, shoiiing the work of the Inductive and Deductive 
Groups compared, section by section, in the Preliminary Tests 
and in the Tests of Reproduction (positive marks only). 



Group A (Deductive). 

Marks for , Average Marks — 

three No. Pre- First Second 

preliminary of liiuinary repro- repro- 

tests. cliildreu. tests. duction. duetion. 

Over 50 18.4 20.8 2«.5 

40 to .50 fi 15.6 27.7 26.5 

30 to 40 5 11.7 25.0 22.6 



Third 
repro- 
duction. 

26.0 

25.0 

23.0 



Group B (Inductive). 

Marks for , Average Marks — 

three No. Pre- First Second 

preliminary of liminary repro- repro- 

tests. children, tests. duction. duction. 

Over 50 6 18.4 28.8 28.3 

40 to 50 6 15.4 26.5 26.2 

30 to 40 5 11.7 25.8 24.8 



Third 
repro- 
duction. 

26.8 

25.2 

25.2 



It seems clear that in this case the children taught 
inductiv^ely were just as successful as those taught 
deductively, even in immediate reproduction, and 
that after a. month's interval they were rather more 
so; they had lost less of what they had been taught. 
This will, perhaps, he shown more clearly in the fol- 
lowing tables: 



Table X, shoicing the icork of the two groups compared, in the 
Preliminary Tests, and in the Tests of Immediate and Deferred 
Reproduction (positive marks only). 



Average mark 
for three pre- 
liminary tests. 

Inductive group 15.4 

M. V.'s 2.4 

Deductive group 15.4 

M. V.'s 2.4 



1 


■Average Marks. 


-^ 


First 


Second 


Third 


repro- 
duction, 


repro- 
duction. 


repro- 
duction. 


27.1 
1.8 

26.6 
1.9 


26.5 
1.9 

25.4 
2.2 


25.8 
2.2 

24.8 
2.4 



, Av 


erage Marks. 


\ 


First 


Second 


Third 


repro- 
duction. 


repro- 
duction. 


repro- 
duction. 


26.8 

1.9 

26.3 

1.8 


26.2 
1.8 

25.0 
2.3 


25.2 
2.3 

23.9 
2.0 



SECOND SERIES OF EXPERIMENTS. 65 

Table XI, showing the icork of the tivo groups compared in the 
PreUminary Tests and in the Tests of Immediate and Deferred 
Reproduction (ichen the negative marks have been subtracted 
fioin the positive 7nar}:s). 



Average marlc 
for three pre- 
liminary tests. 

Inductive group 15.4 

M. V.'s 2.4 

Deductive Group 15.4 

M. V.'s 2.4 

The balance of advantage seems even more clearly 
on the side of the group taught inductively.* 

(c) Correlation Between Immediate and Deferred 
Reproduction. 

It seems likely from Table IX, already given, that 
there is considerable' positive correlation between 
the results of inmiediate reproduction and those of 
deferred reproduction. That is to say, the girls who 
are best immediately after teaching and learning are 
Filso the best after an interval, and those who are 
worst immediately after teaching and learning re- 
main the worst after some time has elapsed. But in 
Table IX the children are classified on the basis of 
their marks for the preliminary tests, and this classi- 
fication tends to obscure much of the correlation 
^vhich undoubtedly exists. In the following tables 
the classification is based on the marks obtained in 
the test of immediate reproduction : 



*In Table.s X and XI the difference between the means of the 
;\'ork of the two groups in deferred reproduction is about twice the 
probable error' in each case, even on the assumption that the 
series are not positively correlated. 



66 INDUCTIVE VS. DEDUCTIVE METHODS. 

Table XII, showing the results for Immediate and Deferred Re- 
production compared, of the Inductive and Deductive Groups 
{positive marks only). 

Deductive Group. 

Marks in No. 

immediate of Average Marks in Reproduction. 

retH'oduction. girls. First. Second. Third. 

Over 28 3 29.3 27.3 26.0 

28 5 28.0 25.8 24.8 

27, 26 5 26.2 26.0 24.6 

Below 26 4 23.3 22.5 24.0 

Inductive Group. 

Marks in No. 

immediate of Average Marks in Reproduction, 

reproduction, girls. First. Second. Third. 

Over 28 5 29.4 28.8 27.0 

28 4 28.0 27.8 27.3 

27, 26 5 26.6 25.0 25.2 

Below 26 3 23.0 23.7 22.7 

It is quite obvious, from the foregoing table, that 
considerable positive correlation exists between im- 
mediate and deferred reproduction, but such a table 
gives us no numerical equivalent for correlation. 
The correlation coefficients have, however, been 
worked out, and for the Deductive Group the coeffi- 
cient for the first and second reproduction is + .62 
(probable error .10), and for the second and third is 
+ .58 (probable error .11). 

In the Inductive Group the correlation coefficient 
between the first and second reproductions is + .76 
(probable error .07), and between the second and 
third is -\- .76 (probable error .07). All the figures 
indicate high reliability for the results, and a com- 
parison of the correlation coefficients for the Induct- 
ive and Deductive Groups shows the work of the 
former to be the more consistent. 



SECOND SERIES OF EXPERIMENTS. 



67 



(d) Results of the Test on New Material. 

In the case of the previous school we found that 
with younger children of a lower standard the de- 
ductive method seemed the better for purely repro- 
ductive purposes. In this school the inductive method 
seems better, even for purposes of reproduction. 
We have now to see whether, when application is 
made to new material, the results for these children 
agree with or differ from those of the preceding 
school. First let me show the results for the two 
groups as wholes : 



Table XIII, sJiowing the work of the two groups compared in the 
Preliminary Tests and in the Tests of Application to Ncio 
Material. 







Average INIarks for New 








Material. 








When 




Average mark 




negative 




for throe 


Positive 


marks 




preliminary 


marks 


have been 




tests. 


only. 


subtracted, 


Inductive group. . . 


15.4 


25.5 


24.2 


M. V.'s 


2.4 


2.7 


3.4 


Deductive gi'oup. . . 


15.4 


23.3 


21.9 


M. V.'s 


2.4 


2.9 


2.9 



We have a clear advantage, in both cases, on the 
side of the Inductive Group. The difference between 
the averages amounts to about three times its 'prob- 
able error,' even on the assumption that the series 
are not positively correlated. Once again, then, we 
liiid the inductive method triumphant when applica- 
tion is made to new material. Let me now show how 
far this is a difference which is to be found all along 
the line, i. e., for the weaker as well as for the abler 
pupils : 



68 



INDUCTIVE VS. DEDUCTIVE METHODS. 



TaJjle XIV, showing the icork of the tioo groups compared, section 
hy section, in the Preliminary Tests and in the Test of Appli- 
cation to New Material (positive marJcJ, and positive marks 
after deduction of the negative marks). 

Group A (Deductively Group B (Inductively 







Taught). 




Taught). 






Marlvs for New 




Marks for New 


Marlis for 




Material. 




Material. 


three 


No. 


(Posi- (After 


No. 


(Posi- (After 


preliminary 


of 


tive de- 


of 


tive de- 


tests. 


girls. 


onlv). duetion). 


girls. 


ouly). duetion). 


Over 50 


(J 


25.8 24.5 





27.3 26.0 


40 to 50 


6 


23.7 21.7 


6 


24.2 22.7 


30 to 40 


. . . . 5 


19.8 19.2 


5 


25.0 24.0 



There seems little doubt that the group inductively 
taught shows a superiority which is general — a supe- 
riority which, somewhat unexpectedly to me, how- 
ever, seems most clearly marked in the weakest (ini- 
tially considered) of the three sections into which 
each group is divided. 



VI. THIRD SERIES OF EXPERIMENTS. 
1. General Plan. 

As in the previous experiments, a whole class of 
children, working under the same teacher, with the 
same curriculum, and according to the same time- 
table of work, was divided into two equal groups on 
the basis of several tests in geometrical defmition, 
which the children attempted without instruction and 
without help. Then, subsequently, one of the two 
groups was taught inductively and the other group 
learnt the definitions. There were tests of immedi- 
ate reproduction immediately after the lesson, and 
another test, which might also be called a test of im- 
mediate reproduction, on the following day. About 
a week later there was a test of application to new 
material, and, three weeks after this, two further 
tests were given, which will be referred to as tests of 
deferred reproduction. 

The work was done with fifty children, whose av- 
erage age was 9 years 3 months. They were graded 
as Standard III of a municipal elementary school for 
boys, situated in a very good suburban neighborhood 
in the southeast of London. The inductive teaching 
was done in this case not by me, but by the teacher of 
the class; whilst the group which studied the written 
definitions was taken, during that particular lesson, 
by the head master of the school. All the tests were 

G9 



70 INDUCTIVE VS. DEDUCTIVE METHODS. 

administered by the class teacher, who had had some 
experience of research work in biology as well as in 
experimental pedagogy. One of the boys' fathers 
told him he was doing Euclid (which he wasn't), and 
gave him a 'tip' or two which affected some of his 
papers adversely; but, with that exception, the suc- 
cess of the experiment was not hindered by any pre- 
vious knowledge on the part of the children. 
Whereas, with the Standard V class of girls, in the 
experiment just described, the teacher's methods 
were instructional rather than either definitely in- 
ductive, deductive, or memoriter, and with the Stand- 
ards VI and VII class of girls, in the experiment 
which has just been described, the teacher's methods 
were both inductive and memoriter, according to the 
subject-matter dealt with; in this third case the re- 
action against unintelligent teaching had gone so 
far that, whilst the inductive teaching was extremely 
good, the memoriter work was decidedly novel to the 
children. Novelty has a stimulating influence, we all 
know, but it is unlikely that its influence is more ef- 
fective in result than that of habitual practices. In 
any case it is essential to try the experiment with 
classes differently taught. 

2. The Preliminary Tests and the Method of 
Marking. 

Just as before, drawings of squares, triangles, ob- 
longs and diameters of circles, with their names writ- 
ten against the drawings, were shown to the children ; 
the questions, "What is a square!" etc., were written 
on the blackboard; the children were told to look at 



THIRD SERIES OF EXPERIMENTS. 71 

the squares, triangles, etc., and to answer tlie ques- 
tions in writing as well as they could. 

The units of marking, as before, were obtained 
after a careful review of the answers actually given, 
and it was found that the units previously adopted 
were quite suitable. A few instances of the children 's 
attempts at spontaneous definition may be worth 
quoting. It must be remembered that these children 
were considerably younger than either of the classes 
of girls whose work has previously been described, 
and that they were graded as Standard III as com- 
pared with Standards V, VI and VII. On the other 
hand, the school was much more favorably situated 
socially than either of the schools for girls. More- 
over, it was a boys' school; and boys, whether 
through greater natural ability or more training, are 
more proficient, geometrically, than girls. 

R. D., aged 9 years 1 month, wrote : 

1. A square is a four sided figure with four points and the sides 
are all equal. 

2. A triangle is a three sided figure with three points and the 
sides equal. 

3. An oblong is a four sided figure with two sides long and two 
sides short. 

4. A diameter is a strait line that goes anything like a circle 
and will go across any way. 

If we mark this paper — R. D.'s first preliminary 
test — on the system of marking adopted in the pre- 
vious experiments,* we see that for his definition of 
a square he receives one mark for 'figure,' one for 
the adjective 'sided,' one for the numerical adjective 
' four, ' and one for the equality of the sides. ' Points ' 
are taken as equivalent to angles or corners, and 



*The reader is recommended to turn to page 28 for the list of 
units. 



i'l INDUCTIVE VS. DEDUCTIVE METHODS. 

therefore receives a mark, whilst the numerical ad- 
jective 'four' also scores. This gives a total of six 
marks for the definition of the square. 

The definition of triangle receives one mark for 
'figure,' one mark for 'sided,' one mark for 'three,' 
one for 'points,' and another for 'three' (points). 
'The sides equal' receives a mark as a 'bad error,' 
but there were so few of these in the preliminary 
tests that they were not tabulated. 

The boy's definition of oblong receives a mark for 
'figure,' one for 'sided,' one for 'four,' one for 'two 
sides long,' and another for 'two sides short.' 

His last definition is rather weak. He obtains a 
mark for 'hue' and one for 'strait,' and that is all. 

When one remembers that these are untaught, 
spontaneous definitions given by a boy 9 years of 
age, we shall, I am sure, regard them as affording 
evidence of considerable ability. Four times the boys 
answered these questions without help and without 
criticism, and advanced a little each time. This is 
what K. D. wrote on his fourth attempt — the fourth 
preliminary test — three days after the first: 

1. A sqnai'e is a four sided figure ^yitll four equal sides and 
four sliai'p points. 

2. A triangle is a three sided figure with three equal sides to it 
and it has three sharp points, 

3. An oblong is a four sided figure with four points but the 
sides are not all the same two sides one length and the other sides 
another length. 

4. The diameter of a oirele is a line that is going from one side 
to the other side of the circle and that is called the diameter of a 
circle and the line is quite strait. 

Let US see how far this fourth paper is in advance 
of the first. The definition of a square receives the 
same mark as before; it is slightly more concise in 



THIRD SERIES OF EXPERIMENTS. 73 

expression, but the units of correct description are 
the same in number in both cases. 

The definition of triangle remains unaltered. 

It is interesting and important to notice that even 
a clever boy may go on perpetrating a 'bad error' 
unless his attention is drawn to it, which, of course, 
the conditions of the experiment did not permit us 
to do in these tests. 

That E. D. is clever for a nine-year-old boy is 
clearer from his next two definitions than from those 
of the square and triangle. He nearly doubles his 
previous mark for his definition of an oblong. He 
now receives marks for 'points,' for 'four' (points), 
for 'two sides long' and for 'two sides short,' and 
also for 'two long sides equal' and 'two short sides 
equal.' 

His definition of diameter has also improved. He 
has now included the condition that it must go from 
one side of the circle to the other. 

These papers of R. D. are extremely good ones, 
and do not represent the average mark of the class, 
which ranges from 11 to 13 units, rather than from 
18 to 23, which R. D. obtains for his first and fourth 
papers, respectively. 

Let me now give examples of some papers below 
the average. 

J. C, aged 9 years 2 months, answered his first pre- 
liminary test as follows : 

1. A square is four put into one shape with equal sides. 

2. A triangle is a thing that has no equal sides, two are equal 
and one is not, and it has three sides. 

3. An oblong is not a square, but it is a long one. 

4. The diameter is a line drawn through the midle of a circle. 

Side by side with this — J. C.'s first preliminary 



74 INDUCTIVE VS. DEDUCTIVE METHODS. 

test — let us compare the paper worked by him three 
days later — his fourth preliminary test : 

1. A square is a shape of a block with four equal sides. 

2. A triangle is a long square with only three sides, the two side 
ones are both the same and the top one is not. 

3. An oblong is a square that is long, with two equal sides and 
two ends whicli are not the same size. 

4. The diameter of a circle is a line drawn down the midle. 

The marks for the definition of a square are in 
both cases the same: 'shape' receives a mark, 'sides' 
receives one, 'four' gets one, and 'equal' (sides) gets 
one. 

The two definitions of a triangle receive the same 
mark: there is a mark for 'sides' and one for 'three' 
(sides), and that is all. 

The first definition of an oblong receives no marks 
at all, whilst the one given later receives a mark for 
'sides,' a mark for 'two equal' (sides), and one for 
' ' two ends which are not the same size as the others. ' ' 

His definition of a diameter remains unchanged 
throughout the preliminary tests ; in each case it re- 
ceives two marks only, one for 'line' and one for 
' drawn through the middle. ' 

J. C. advances from a mark of 8 in the first test to 
11 in the fourth.* 

Having given some indications of the work done in 
the Preliminary Tests, on the results of which the 
class was divided into two equal groups, let me set 
out in detail the chronological progress of the experi- 
ment. 



*The average improvement from test to test is shown on page 91. 



THIRD SERIES OF EXPERIMENTS. 75 

3. Chronology of the Experiment. 

A first Preliminary Test was given at 9.40 A. M. 
on Monday, October 23, 1911, immediately after 
Scripture lesson; a second on Tuesday, October 24, 
a third on Wednesday, October 25, and a fourth on 
Tlmrsda)^, October 26, at the same hour and after the 
same lesson on each occasion. On the results of these 
four tests the class was divided into two equal 
groups. 

On Thursday, November 9, at 9.40 A. M., one of 
the two groups was taught the definitions inductively 
by the methods already described, whilst the other 
studied them as written out, with reference to the 
drawings appended to the verbal descriptions. 
Twenty-three minutes were taken by the teacher to 
teach the definitions inductively ; the same time was 
allowed to the group which was studying the defini- 
tions with a view to remembering them. Both 
groups of children were aware that they were to be 
tested on their work at the close of the lesson. Ac- 
cordingly, at 10.15 A. M., a test was given in immedi- 
ate reproduction. In this school, since the children 
were young and the exercises very novel, we thought 
it best to take another test, identical with the test of 
immediate reproduction, at the same hour on the 
next day, Friday, November 10, to see how far the 
first day's test was reliable. These two tests will be 
referred to as the two Tests of Immediate Eepro- 
duction. 

At 9.40 A. M. on Thursday, November 16, a test 
was given on the application of what had been learnt 
to new analogous material with the object of discov- 



76 INDUCTIVE VS. DEDUCTIVE METHODS. 

ering which of the two groups attacked the new mate- 
rial the more successfully. 

Finally, two tests of deferred reproduction were 
given at 9.40 A. M. on Thursday, December 7, and 
Friday, December 8. The children were quite una- 
ware that they would be required to take any of these 
tests, with the exception of the one immediately after 
the teaching and learning on Thursday, November 9. 

4. The Final Tests and the Method of Marking. 

The two tests of Iimnediate Reproduction were 
repetitions of the Preliminary Tests, as were also 
the two tests of Deferred Reproduction. The units 
of marking previously used in the Preliminary Tests 
were found quite satisfactory. The tests of deferred 
reproduction received negative as well as positive 
marks. One or two specimens of the worked papers 
may be of interest. 

L. 0., aged 9 years, who scored 13, 16, 18, 18 in his 
preliminary tests, and was taught inductively, for his 
first test of Immediate Reproduction on November 
9, wrote as follows : 

1. A square is a shape with four lines all the same size and for 
corners all the same size. 

2. A triangle is a shape with three corners and three lines. 

3. A oblong is a shape with two long lines the same size, and 
two shorter lines the same size. 

4. A diameter of a circle is a line which goes from one part to 
the opposite part touching the middle of the circle and keeps inside 
the circle. 

This is a good set of answers for a boy only nine 
years of age. Marked on the system of units previ- 
ously used, the definition of a square receives seven 
marks, the definition of a triangle receives five 



THIRD SERIES OF EXPERIMENTS. 77 

marks, that of the oblong receives seven marks, and 
that of the diameter of a circle receives three marks. 
It will be seen that, compared with the standard defi- 
nitions, there is a loss of one mark in the definition 
of a square, since the description 'straight' is not 
applied to the 'lines' or 'sides.' The definition of 
triangle is correct. 

Six marks are lost on the oblong. ' Four equal cor- 
ners ' are omitted, carrying three marks. 'Straight' 
is omitted in describing the lines or sides, and the two 
long lines and the two short lines are not described 
as opposite. 

One mark only is lost on the definition of 'diam- 
eter;' the line is not described as 'straight.' The 
marks, totaled, amount to 22. 

On the next day's test L. 0. goes down one mark. 
His definitions of square and triangle remain un- 
changed. In his definition of oblong he omits the two 
points previously inserted, namely, that the two long 
lines are of the same length, and that the two short 
lines are of the same length. But in the definition 
of the diameter of a circle he inserts the description 
'straight' which he had omitted the day before alto- 
gether. He thus scores 21 marks for his second test. 

Let us now see what happens a month later when 
the same test is applied a third time. I give his 
paper in full. 

L. 0., aged 9 years 1 month, on December 7, 1911, 
in his first test of Deferred Eeproduction, wrote as 
follows : 

1. A square is a shape with four lines all the same length, and 
four corners all the same size. 

2. A triangle is a shape with three lines and three corners. 

3. An oblong is a shape with four lines two long lines both the 
same size, and two shorter lines both the same length. 



78 INDUCTIVE VS. DEDUCTIVE METHODS. 

4. A diameter of a circle is a line inside which goes from one 
part of the circle and touches the middle of the circle goes on to 
the opposite part of the circle to where it started and it must be a 
straight line. 

This is a very good paper, and scores a total of 23 
marks, an advance on the work of the raonth before. 
On the day following, on which was given the second 
test of Deferred Reproduction, L. O. scored 24 
marks, for the description 'straight' of the sides of 
the square, omitted on December 7, was included on 
December 8. His average mark for his two tests of 
Immediate Reproduction, those, namely, of the 9th 
and 10th of November, was 21.5; his average mark 
for his two tests of Deferred Reproduction was 23.5. 

It must not be thought that every boy obtains more 
marks a month after the lesson than he does for his 
immediate tests, but many of them do; and the aver- 
age result shows only a slight decline, rather more 
marked in the group taught deductively than in the 
group taught inductively. This is explained by the 
fact that both the teaching and the learning were well 
within the comprehension of the boys. When this is 
the case, and they work in consequence with interest 
and enthusiasm, they forget surprisingly little. 

It may now be of some value if I give the corre- 
sponding papers of a boy in the Deductive Group. 

R. S., aged 9 years 2 months, who scored 14, 19, 19, 
18 marks in his four preliminary tests, in his first 
test of Immediate Reproduction wrote : 

1. A square is a shape with four sides and four corners. The 
sides are straight and all the same length. The corners are all the 
same size. 

2. A triangle is a shai)e with three sides and three corners. 

3. An oblong is a shape with four sides and four corners. The 
sides are straight and there are two long sides and two short sides. 



THIRD SERIES OF EXPERIMENTS. 79 

The long sides are opposite oue another and are the same length, 
and the two short sides are opposite and are the same length. 

The diameter of a circle is a straight line which goes through 
the centre of the circle. 

Only two units of definition are omitted : the equal- 
ity of the angles is left out in the definition of the 
oblong, and the delimitation of the diameter by the 
opposite parts of the circumference of the circle is 
omitted in the last definition. It is an excellent pa- 
per, appearing on the face of it, if one compares it 
with the verbal expression of the definitions which 
were given to be studied, to owe a great deal to a 
highly developed rote memory. If that is so, it is 
memory for statements that are really understood, 
since they persist unchanged without the subsequent 
intrusion of stupid errors, and an unusually high 
mark is obtained by this boy for his power of appli- 
cation to new material. I propose to defer consid- 
eration of the latter issue, since just now we are con- 
cerned only with the tests of Immediate and De- 
ferred Reproduction. 

The next day R. S. obtained 29 marks, as compared 
with 28 of the previous day. There were slight 
changes of verbal expression. For instance, the tri- 
angle became "a three cornered figure with three 
sides." The equality of the angles was omitted in 
the square, but on this occasion, though not in the 
previous test, was included in the definition of the 
oblong. In the definition of the circle an improve- 
ment was shown ; the point was included which 
the day before had been omitted; it was now men- 
tioned that the line went from ''one side to the oppo- 
site side of the circle." 

One month later R. S. scored 28 marks. He 



80 INDUCTIVE VS. DEDUCTIVE METHODS. 

omitted the description 'straight' in his definition of 
a diameter of a circle which he had before included. 
On the day following he made the same omission. 
Otherwise his definitions are just as good as those 
which he had given a month before. His average 
mark for Immediate Reproduction is 28.5, and for 
Deferred Reproduction is 28.0. 

Let me give one more illustration, the work of a 
boy who obtained 6, 6, 7 and 7 marks in his four pre- 
liminary tests, and who also was taught in the De- 
ductive Group. 

H. W., aged 10 years 1 month, in his first test of 
Immediate Reproduction wrote : 

1. A square is a shape with four corners aud four sides the 
same size. 

2. A triangle is a shape with three corners and three sides. 

3. An oblong is a shape with two small sides, and two big sides 
opposite one another. 

4. The diameter of a circle is a line passing through the middle 
of the circle. 

Marked on the same units as before, the definition 
of a square obtains six marks; the definition of tri- 
angle obtains five marks ; the definition of oblong re- 
ceives six marks, for it is called a 'shape,' its 'four' 
'sides' are implied, its 'two long sides' and its 'two 
short sides' are noted, and the fact that its 'two long 
sides are opposite each other.' The definition of 
diameter receives two marks. This is not a strong 
paper; it scores 19 marks only as a total, but it im- 
plies a very great advance on this boy's preliminary 
tests. One point of interest lies in this. Whereas, 
in the papers of R. S., recently given, there was an 
appearance of rote learning in the answers, there is, 
in the case of this boy, no direct indication of that. 

H. W.'s next test of Immediate Reproduction, 



THIRD SERIES OF EXPERIMENTS. 81 

worked on the following day, receives the same num- 
ber of positive marks, namely, 19. A 'bad error' has 
crept in, for the corners of the triangle are described 
as all the same size. The 'two small sides' of the 
oblong are now called "two small tops," but this and 
the 'bad error' are the only changes. One month 
later, for his first test of Deferred Reproduction, 
H. W. wrote : 

1. A square is a shape with four corners and four sides opposite 
one another and they are all of the same length. 

2. A triangle is a shape with three corners and three sides they 
are not opposite one another. 

3. An oblong is a shape with three corners and three sides, it is 
a zig-zag shape not all the same length. 

4. A diameter of a circle is a line passing through the middle 
of it. 

Considerable changes are evident in this paper. 
There are, as before, six positive units of correct 
description in the definition of the square; but the 
statement "four sides opposite one another" has 
been adjudged a 'bad error.' It is, of course, the 
confused application of some phrase remembered, 
but not understood. Let it not be supposed, however, 
that no child inductively taught makes similar errors. 

The definition of triangle receives five marks as 
before. The memory of the oblong has largely gone. 
It is still remembered that it is a 'shape' and has 
'corners' and 'sides,' and thus three positive marks 
are obtained. But to give an oblong ' three ' corners 
and 'three' sides and to call it 'zig-zag' shape is held 
to involve three bad errors. The definition of diam- 
eter remains unchanged, and scores two marks. The 
paper as a whole receives 16 positive marks, and 
there are four marks for bad errors. 



82 INDUCTIVE VS. DEDUCTIVE METHODS. 

But on the next clay, in his second test of Deferred 
Eeproduction, H. W. made a decided recovery. He 
then wrote : 

1. A square is a shape with four corners and four sides they 
are opposite one another, with all the sides and corners an equal 
size. 

2. A triangle is a shape with three sides and corners it is a 
zig-zag shape. 

3. An oblong is a shape with four sides, two long sides and two 
short tops they are opposite one another. 

4. A diameter of a circle is a line passing through the middle 
of it. 

This is undoubtedly H. W. 's best paper. He scores 
the highest marks he has yet scored for the definition 
of the square, namely, seven positive marks, since, 
for the first time, he has mentioned the equality of 
the corners, but he retains liis 'bad errors.' The defi- 
nition of a triangle remains unchanged in correct 
units ; it is held inadmissible to call the triangle a zig- 
zag shape. The definition of oblong has returned to 
its first condition; indeed, it is rather better, for it 
is easier now to regard H. W. as implying that the 
'two long' and 'two short' sides are pairs of equals. 
The mark for the double equality is, however, not 
given, as the meaning is somewhat doubtful. The 
definition of diameter remains unchanged. H. W. 
scores 20 positive marks for his paper and one nega- 
tive mark for a 'bad error.' Again we find the marks 
for Deferred Reproduction not much inferior to 
those of Immediate Reproduction in this case; in- 
deed, the last paper is the best the boy did through- 
out the series. 

I trust that the inclusion of these papers will be of 
service in giving life and body to the rather bloodless 
array of figures, which I give subsequently, dealing 



THIRD SERIES OP EXPERIMENTS. 83 

with the results of the tests in Immediate and De- 
ferred Reproduction. 

The Test of Application to New Material was iden- 
tical witli that used in the experiment j^reviously de- 
scribed. Drawings of rhombuses, trapeziums, rhom- 
boids and diagonals of squares, with their names 
appended, were shown to the children, and they were 
required to answer in writing the questions: "What 
is a rhombus?" etc. It may add to the facility with 
which the progress of this experiment is understood 
if I give verbatim one or two of the worked papers. 
In the test of application to new material negative 
marks were assigned as well as positive marks. 

L. 0., aged 9 years, a boy who worked in the In- 
ductive Group, whose work in Immediate and De- 
ferred Reproduction has already been quoted, wrote 
the following paper in this test : 

1. A rhombus is a shape contaiiiiug four lines all the same 
length, so that if you loolved at it one \A'ay it seems to bend back- 
ward, and if you look at it again it looks to bend forward. 

2. A rhomboid is a shape also containing four lines, two long 
lines both the same length, and two shorter lines both the same 
lengtli. 

9. A trapezium is a shape with four lines three long ones, and 
one short one. 

4. A diagonal of a square is a streight line going from one cor- 
ner to its opposite one. -^ 

L. 0. receives four marks for his definition of 
rhombus — one for 'shape,' one for 'lines,' one for 
'four,' and one for all the same 'length.' He receives 
three marks for his definition of trapezium, one for 
'shape,' one for 'lines,' and one for 'four.' His 
statement that there are three long lines and one 
short one was not held to be equivalent to the state- 
ment that the sides were unequal, but it was not con- 
sidered a ' bad error. ' For his definition of rhomboid 



84 INDUCTIVE VS. DEDUCTIVE METHODS. 

he obtains seven marks — one for 'sliape,' one for 
'lines,' one for 'four,' one for 'two long- lines,' one 
for 'two shorter lines,' and two for the pair of equal- 
ities in the length of the lines. 

The definition of diagonal receives four marks, one 
for 'line,' one for "streight," and two for "from one 
corner to its opposite one. ' ' 

The paper scores a total of 18 positive marks, and 
there are no 'bad errors ;' the average mark obtained 
by the boys of the Inductive Group is rather lower 
than this. 

R. S., aged 8 years 5 months, who also was taught 
in the Inductive Group, wrote : 

1. A rhombus is a figure with four straight sides and four 
equal corners. 

2. A trapezium is a figure with four corners which are equal 
with four sides. 

3. A rhomboid is a figure with two small sides which are hori- 
zontal and two bigger parlerlell lines equal. 

4. A diagonal of a square is a straight line from one corner to 
another corner. 

The first definition receives a mark for 'figure,' a 
mark for 'sides,' one for 'four,' one for 'straight,' 
one for 'corners,' and one for 'four;' a total of six 
positive marks; but there is one 'bad error' — the 
corners are not equal: boys taught inductively can 
obviously make the same sort of blatant error as 
boys taught deductively. 

The second definition receives a mark for ' figure, ' 
one for ' corners, ' one for ' four, ' one for ' sides, ' and 
one for 'four' (sides) ; a total of five positive marks ; 
but, again, there is a 'bad error' — the angles of the 
trapezium are not equal. The definition of rhomboid 
obtains seven positive marks — one for 'figure,' one 
for 'sides,' one for 'two small' (sides), one for 'two 



THIRD SERIES OF EXPERIMENTS. 85 

bigger' lines, a mark for saying the two bigger are 
equal, and one for saying the two bigger lines are 
parallel. There is one 'bad error;' the two small 
sides were in one case only drawn horizontally. The 
definition of diagonal receives three positive marks 
— one for 'line,' one for 'straight,' and one for "from 
one corner to another : ' ' the further specification of 
'opposite' corner is omitted. 

The paper, as a whole, gains 21 positive marks, 
with three negative marks for ' bad errors. ' 

Let me quote one more illustration from among 
the boys who were taught inductively. 

H. B., aged 9 years 2 months, wrote : 

1. A rhombus is a fugare which is like a square and has fore 
corners. 

2. A trapezium is something like a triangle only it has fore 
corners. 

3. A rhomboid is a sought of fugare which is something like 
an oblong. 

4. A diagonal of a square is a square with a line across the 
midal. 

This is a very weak paper ; it was worked by a boy 
who was almost at the bottom of the Inductive 
Group in the preliminary tests, and he seemed to jus- 
tify his position. It is psychologically interesting 
that he apprehended the similarity between the work 
now required and the work he had been taught, but 
was unable to specify the points of similarity and 
ditference between the figures of the first set and the 
figures of the second set. He had but little know- 
ledge and could not apply much of that. His marks 
are: three for his definition of rhombus, two for his 
definition of trapezium, one for his definition of 
rhomboid, and one for the definition of the diagonal 
of a square. "Across the midal" is not held to be 



86 INDUCTIVE VS. DEDUCTIVE METHODS. 

wrong, though it might be ; in any case it is not re- 
garded as sufficiently definite to obtain a mark. It 
is regarded as a 'bad error' to say that the diagonal 
of a square is a square. H. B. receives a total of 
seven positive marks, with one negative mark for 
bad errors. 

It is, perhaps, worthy of note that this boy falls 
from 17 in his test of Immediate Reproduction to 
9 in his test of Deferred Reproduction. He can- 
not apply his old knowledge and he cannot remember 
it for more than a day or two. 

Let us now turn to some illustrative examples of 
the work of the group taught deductively. 

G. M., aged 8 years 1 month, wrote : 

1. A rbombus is a figure with two slanting sides and two 
straight ones arranged so that two of the sides are facing each 
other and the other two opposite each otlier and also four corners. 

2. A trapezium is a figure with four slanting sides arranged so 
that there are two sides nearly the same length, these two are 
generally touching each other. Then there is a smaller one and 
yet a smaller one still, so that there are four sides and two equal 
ones the others ofcourse are not. 

3. A rhomboid is a figure with two slanting sides and two 
straight ones and also four corners two of the sides are longer 
than the other two and also are opposite one another and so are the 
two shorter sides. There can be ones upright and lying down and 
also slanting ones. 

4. A diagonal of a square is a line drawn from one corner to 
the other it need not have to be drawn from a corner for it could 
be from the middle of the top to the middle of the bottom, but you 
can't have it so that it is from the middle of the one side to the 
middle of the bottom or to the middle of the top. For the diameter 
is the greatest and longest line you can have across it or down it 
and that wouldn't be the longest, not nearly. 

This is an excellent paper for a boy of eight years 
of age. He was taught in the Deductive Group, but 
evidently he is quite capable of applying what he has 
learnt. It would be a serious error to suppose that 
because a boy has learnt a set of definitions therefore 



THIED SERIES OF EXPERIMENTS. 87 

he cannot apply tliem. In a very large number of 
cases he certainly can. The contention raised in this 
monograph is that inductive teaching produces a 
higher transfer to new material than deductive, not 
that deductive teaching involves no transfer at all. 
This first-rate paper may do something to prevent 
an exaggerated conclusion which the subsequent fig- 
ures may not succeed in adequately moderating. Let 
us mark the paper on the usual system of units. G. 
M. is evidently using the word straight to mean, as it 
often does with boys, horizontal and vertical ; he does 
not mean that only two of the lines are ' straight ' in 
the proper sense. And he is wrong on his own mean- 
ing, for one of the rhombuses drawn had neither ver- 
tical nor horizontal lines, but two of them had, and 
to these he has apparently confined his attention. 
He receives a mark for * figure,' a mark for 'sides,' 
and one for 'four' (sides), which is involved in his 
pair of twos, and one for 'corners' and one for 'four.' 
He gets two marks for seeing that the opposite sides 
are paired. This marking yields a total of seven pos- 
itive marks, whilst he receives a negative mark for 
being wrong on his own meaning of 'straight.' In 
his definition of trapezium he receives a mark for 
'figure,' one for 'sides,' and one for 'four.' His first 
description of the sides is held to be equivalent to 
saying they are unequal, so he receives a mark for 
that. Later he is marked for a 'bad error' in saying 
that two of the sides are equal. They are so in one 
of the trapeziums only. For the definition of trape- 
zium, then, he gets four positive marks, with one neg- 
ative mark for a 'bad error.' Again, in his definition 
of a rhomboid we find a misuse of the word straight, 
and again he is wrong, even on his own meaning. 



88 INDUCTIVE VS. DEDUCTIVE METHODS. 

But be obtains positive marks for 'figure,' for 'sides,' 
for 'four' (sides), for 'two long' (sides), for 'two 
shorter' (sides), and two marks for noting tbe pairs 
of opposites. He also notes tbe 'four corners.' He 
tbus receives nine positive marks and one mark for a 
'bad error.' 

His definition of diagonal is extremely interesting. 
He receives two positive marks only — one for 'line' 
and one for "from one corner to anotber." After 
tbat, alas! tbe transfer from diameter (tbe corre- 
sponding definition wbicb was learnt) bas been too 
tborougb. No diameters were drawn in tbe squares 
wbicb were before tbe boy's eyes, and it is not unfair 
to call tbe lapse into diameter a 'bad error.' Tbis 
definition receives therefore two positive marks and 
one negative mark. Tbe paper, as a wbole, receives 
a total of 22 positive marks, and tliere are four bad 
errors; it is considerably above tbe average of tbe 
papers worked by tbe Deductive Group generally. 

H. W., aged 10 years 1 month, whose work in Im- 
mediate and Deferred Reproduction has already 
been quoted, wrote tbe following in bis test of appli- 
cation to new material : 

1. A rhombus is a shape something like the shape of a diamond. 

2. A trapezium is a shape with four corners not opposite one 
another their are different shapes of trapeziums they are a zig- 
zag shape some corners longer than others, they are not squares. 

3. A rhomboid is a shape with two small tops both opposite one 
another, and with two long sides with the corners exactly opposite 
one another. 

4. A diagonal of a square there is a square and a line passes 
right through. Sometimes they pass from side to side Qther times 
from corner to corner. 

H. W.'s definition of rhombus receives one mark 
only — a mark for tbe description 'shape.' For tbe 
definition of traiDezium three positive marks are 



THIRD SERIES OF EXPERIMENTS. 89 

gained — one for 'shape,' one for 'corners,' and one 
for 'four.' There are no 'bad errors.' It was not 
thought admissible to regard the expression "some 
corners longer than others" as involving the ine- 
quality of the angles. His definition of a rhomboid 
receives a mark for 'shape,' one for 'sides,' one for 
'corners,' and one for 'four' sides, for the number of 
sides is involved in the rest of his answer. He also 
receives a mark for "two small tops," one for "two 
long sides," and one for noting that the two small 
sides are 'opposite' each other. The opposition of 
the angles has not been allowed for in the system of 
marking. This definition therefore receives a total 
of seven positive marks. The definition of diagonal 
receives two positive marks only — one for 'line' and 
one for 'from corner to corner.' It was regarded as 
a bad error to say that "sometimes they pass from 
side to side. ' ' The total marks for this paper amount 
to 13 positive marks, from which one has to be de- 
ducted for ' bad errors. ' 

Let me now pass to the work of a boy who scored 
13, 12, 12 and 11 in his four preliminary tests. It 
seems likely from these figures that we are dealing 
with a boy of little educability, and this suggestion is 
confirmed by his later work.. In his two tests of Im- 
mediate Reproduction he scores an average of 18.5 
marks; in both tests of Deferred Reproduction he 
scores 13 marks, so that a month afterwards he is 
back again to the position he occupied before he 
learnt the definitions, and he completely fails in ap- 
plying what he has learnt. 

A. R., aged 8 years 6 months, the boy whose work 
has just been described generally, wrote : 

1. A rbouabiis is a square which is not strate up. 



90 INDUCTIVE VS. DEDUCTIVE METHODS. 

2. A trapezium is a four-sided thing wliicli sides are not all 
strate. 

3. A rboiuboid is lilve an oblong but its lines are not strate up. 

4. A diagonal of a squai'e is a diameter of a circle only it is 
a squear. 

A. R. has seen some general resemblance between 
the 'figures' of his first set of definitions and those of 
his second set, bnt the resemblances have hindered 
rather than helped him, for a rhombus is not a 
square, and a diagonal of a square is 7iot a diameter 
of a circle. The meaning of the word ''strate" is 
misconceived ; his reference to the sides of the trape- 
zium is not incorrect on the basis of his own mean- 
ing. Of positive marks, on the system of marking 
adopted, he can obviously obtain very few. He 
scores no marks for his definition of rhombus, two 
for his definition of trapezium, one for his definition 
of rhomboid, and none for his definition of diagonal. 
His three positive marks are subject to a deduction 
of two for the 'bad errors' previously indicated. 
Boys of this kind are the despair of the teacher, but 
the evidences ^delded by his work do not point so 
much to stupidity as to ineducability. 

Possibly the reader may already have gathered 
from a perusal of the papers which I have used as 
illustrations some opinions of his own as to the rela- 
tive applicability of the two methods of teaching and 
learning. But all such opinions need to be confirmed 
or modified by a consideration of the tables of results 
which are set out in the next section. 

5. Results of the Experiments, 
(a) Results of the Preliminary Tests. 

The marks for the four preliminary tests were 
fairly steady, decidedly so, when the age of the chil- 



THIED SERIES OF EXPERIMENTS. 91 

dren was taken into consideration, A'^ery few of the 
boys made any violent jumps, and there was a gen- 
eral improvement from exercise to exercise. 

In the first test the average mark was 11.1, in the 
second 12.3, in the third 12.9, and in the fourth 13.1. 
The correspondences between the results of the first, 
second, third, and fourth Preliminary Tests are 
shown compendiously in the following table : 

Table XV, shoicing the correlation beticecn the results of the four 
Preliminary Tests. 

Marks in 

the four No. ,— Average Marks in Preliminarj' Tests.— ^ 

preliminary of First Second Tliird Fourth 

tests. boys. test. test. test. test. 

70 and over 4 16.5 19.8 18.3 18.8 

60 to 70 8 14.8 14.4 16.5 17.3 

50 to 60 11 12.5 13.4 13.9 14.3 

40 to 50 15 10.4 11.3 12.1 11.9 

Below 40 12 6.7 7.9 8.9 9.0 

There is obviously high positive correlation be- 
tween the results of the successive preliminary tests. 
The mental functions we are testing appear to be 
working very steadily. Exact numerical values for 
the coefficients of correlation have been worked out 
from the 50 individual cases on the Pearson formula. 
Between the results of Tests 1 and 2 the correlation 
coefficient is + ,76 (probable error ,04), between 
Tests 2 and 3 is + .79 (probable error ,03), and be- 
tween Tests 3 and 4 is -j- ,80 (probable error ,03). 
These high correlations between the results of the 
successive tests give us reasonable expectations that 
a division into two equal groups may be satisfac- 
torily effected. The boys were divided into two 
equal groups containing 25 children each. The fol- 
lowing table will indicate how the division was made : 



92 INDUCTIVE VS. DEDUCTIVE METHODS. 

Table XVI, showing the Division into Tiro Equal Groups. 

Group A. 

Name 

(Initials , Marks for Preliminary Tests. , 

only). First. Second. Third. Fourth. Total. 

R. D 18 22 18 22 80 

A. C 17 18 19 16 70 

L. 13 16 18 18 Go 

W. G i 9 6 12 31 

G. k .3 5 7 10 25 

Averages 11.1 12.4 12.9 1.3.3 49.7 

M. V.'s 3.5 2.9 2.5 2.4 

Group B. 

Name 

(Initials , Marks for Preliminary Tests. n 

only). First. Second. Third. Fourth. Total. 

H. B 17 20 17 17 71 

R. S 14 19 19 18 70 

C. L 16 17 16 18 67 

S. B. . 9 8 il 5 33 

A. W 4 8 6 6 24 

Averages 11.1 12.3 13.0 12.8 49.3 

M. V.'s 2.7 2.8 2.7 3.3 

The average mark per boy per test for Group A 
was 12.4 (mean variation 2.6), and for Group B was 
12.3 (mean variation 2.6). The average age of Group 
A was 9 years 3 months, and of Group B was also 
9 years 3 months. 

(b) Results of the Tests in Immediate and Deferred 
Reproduction. 

It now remains to be shown which of the two 
groups was the more successful when tested on pre- 
cisely what they had been taught or learnt. 



THIRD SERIES OF EXPERIMENTS. 



93 



First, let me give the marks of the two groups as 
wholes, together with their variability: 

Tabic XVII, slioicing the icork of the Inductive and Deductive 
Groups compared, in the Preliminary Tests and in the Tests 
of Immediate and Deferred Reproduction (positive marks 
only). 



1 


First 


Second 


IKS. 


^ 


For all 


imme- 


imme- 


First 


Second 


four 


diate 


diate 


deferred 


deferred 


preliminary 


repro- 


repro- 


repro- 


repro- 


tests. 


duction. 


duction. 


duction. 


duction. 


Inductive group. . . 12.4 


18.8 


18.6 


18.0 


18.1 


M. V.'s 2.6 


2.6 


2.5 


3.0 


3.2 


Deductive group. . . 12.3 


20.5 


20.6 


18.8 


19.4 


M. V.'s 2.6 


3.4 


4.1 


3.4 


3.5 



In the tests for deferred reproduction, it will be 
remembered, negative marks were given as well as 
positive marks. The marks for the two groups are 
given below after the negative marks have been sub- 
tracted from the positive marks : 

TaWe XVIII, showing the marks (after deduction) for the Induct- 
ive and Deductive Groups compared, in the Preliminary Tests, 
and in the Tests of Deferred Reproduction. 





Average 
mark 


1 Average 

First 


Marks. ^ 

Second 




for four pre- 


deferred 


deferred 


Inductive gi'oup. . 
M. V.'s 


liminary tests. 

12.4 

2.6 


reproduction. 

17.7 

3.2 


reproduction. 

17.8 

3.4 


Deductive group.. 
M. V.'s 


12.3 

2.6 


18.6 
3.6 


19.1 
3.7 



There seems no doubt that, when the tests are given 
on precisely the subject-matter which has been learnt 
or taught, the group which learnt the definitions did 
better work than that which was taught inductively, 
and this is true both in immediate and deferred re- 
production, and for both positive and negative 



94 



INDUCTIVE VS. DEDUCTIVE METHODS. 



marks. This conclusion must, of course, be drawn 
subject to the age and mental proficiency of the pu- 
pils. It now remains to be seen whether the differ- 
ence between the groups is one which is common to 
the more proficient as well as to the less proficient 
pupils : 

Tabic XIX, showing the marks of the two groups compared, section 
by section, in the Preliminary Tests and in the Tests of Imme- 
diate and Deferred Reproduction (positive marks only). 

I Inductive Group. , , Deductive Group. , 

Average Average 

mark of Average mark of Average 

two tests mark of two tests mark of 

Marks imme- two tests imme- two tests 

in four No. diate deferred No. diate deferred 

preliminary of repro- rejiro- of repro- repro- 

tests. boys, duction. duction. boys, duction. duction. 

70 and over. .. 2 19.0 19.2 2 24.0 21.0 

60 to 70 4 21.2 19.5 4 22.5 20.9 

50 to 60 5 19.7 10.2 6 22.9 21.0 

40 to 50 8 18.0 18.5 7 18.8 18.3 

Below 40 6 17.0 14.8 6 17.8 16.0 

It seems clear that there is a balance of advantage 
all along the line in favor of the group which learnt 
the definitions, so far, at least, as the positive marks 
are concerned. It now remains to be shown whether 
this is also true when the negative marks are de- 
ducted from the positive marks : 

Table XX, showing the marks of the two groups compared, section 

by section, in the Prelim inanj Tests and in the Tests of De- 
ferred Reproduction (after the negative marks have been de- 
ducted). 

r-Inductive Group.— ^ ^—Deductive Group.— ^ 

Marks Average Average 

in four No. mark in two No. mark in two 

preliminary of tests deferred of tests deferred 

tests. boys. reproduction. boys. reproduction. 

70 and over 2 19.3 2 21.5 

60 to 70 4 19.1 4 20.5 

50 to 60 5 19.1 6 20.9 

40 to 50 8 18.4 7 18.4 

Below 40 6 14.2 6 15.4 



THIRD SERIES OF EXPERIMENTS. 95 

Again, there seems a decided balance of advantage 
on the side of the group which learnt the definitions 
deductively. 

(c) Correlation Between the Results of Immediate 
and Deferred Reproduction. 

It would seem likely from the tables given above 
that the tests given immediately after the teaching 
and learning may be regarded as fairly significant of 
the relative position of the two groups even after 
considerable time has elapsed — in this case after a 
month. As this is a very important issue for experi- 
mental pedagogy, it may be well to subject the hy- 
pothesis to further determination. The following 
tables will show in a general way how far the sug- 
gestion may be taken as valid : 

Table XXI, showing the correlation heticeen the marks obtained in 
the various Tests of Reproduction {positive marks only). 

Inductive Group. 

Marks for the 
first test of No. Average Marks per Boy in the Repro- 

inimediate of ductive Tests. 

reproduction. boys. First. Second. Third. Fourth. 

Over 25 2 27.0 26.5 23.5 25.5 

20 to 25 3 22.3 19.7 20.7 20.7 

18 to 20 7 19.6 19.7 19.1 19.7 

16 to 18 6 17.7 18.1 16.0 15.7 

15 to 16 4 16.0 15.0 16.0 16.0 

15 and under 3 14.3 15.3 16.0 14.3 

Deductive Group. 

Marks for the 
first test of No. Average Marks per Boy in the Repro- 

immediate of ductive Tests, 

reproduction. boys. First. Second. Third. Fourth. 

Over 28 2 29.0 27.0 25.5 26.0 

25 to 28., 3 26.7 26.0 22.3 22.7 

21 to 25 4 23.5 23.2 22.2 23.2 

17 to 21 8 19.7 20.4 18.5 19.7 

16 to 17 4 17.0 18.7 16.2 15.5 

16 and under 4 13.7 13.2 12.5 12.7 



96 INDUCTIVE VS. DEDUCTIVE METHODS. 

It is obvious that considerable positive correlation 
exists between the results of the successive exercises. 
A more precise determination may, of course, be 
made by means of a correlation coefficient. Worked 
out by the standard formula from the individual 
cases, the following are the coefficients : For the In- 
ductive Group the results of the first Test of Eepro- 
duction correlate with those of the second to the 
extent of + .78, the second with the third to the ex- 
tent of + .57, and the third with the fourth to the 
extent of + .85. For the Deductive Group the corre- 
lation coefficients are : first and second tests, + .86 ; 
second and third tests, + .68 ; third and fourth tests, 
+ .94. 

It is obvious that tests given immediately may be 
fairly regarded as indicative of what will happen 
later on, at least in such exercises as these, when we 
are making comparison of one group with another. 

(d) Results of the Test on New Material. 

We have seen already, when the tests required an 
exact reproduction of what had been learnt or taught, 
that the children in this school who learnt deduc- 
tively, like the Standard V children of the girls' 
school in the experiment first described, did better 
work than the group taught inductively. But in both 
the experiments previously described it was found, 
when the test given was on new material, that the 
children taught inductively did better work than 
those who learnt their definitions. Is this advantage 
also to be found on the side of the inductive group 
in this school? These children are younger and are 
less proficient mentally, according to school grading, 



THIRD SERIES OF EXPERIMENTS. 97 

than either of the girls' classes whose work has been 
described. Moreover, they are boys, not girls. Are 
these variations in conditions such as to produce a 
difference in the results? It will further be remem- 
bered that the inductive group, in this case, was 
taught by its own teacher, and not by me, so that 
any intensity of impression due to personal novelty 
was thereby eliminated. 

Perhaps I may be pardoned for a sentence of ap- 
parent digression. I hold it extremely important for 
the science of experimental pedagogy that no result 
should be taken as valid for general application un- 
less the use of it is justified by its success in the hands 
of the usual teachers of the school. Its success in 
the hands of the specialist or other exceptional per- 
son is quite insufficient to recommend it for general 
adoption. Let us, then, see what the results were 
when the whole experiment was conducted by the 
teachers themselves. I shall show the work of the 
two groups compared both in the Preliminary Tests 
and in the Test of Application to New Material. 
First, let me give the results of the two groups as 
wholes : 

TaT)le XXII, showing the toorJc of the tiro groups compared, in the 
Preliminary Tests and in the Test on Neto Material. 



Average mark 

for four 

preliminary tests. 

Inductive gi*oup 12.4 

M. V.'s 2.6 

Deductive group 12.3 

M. v.'s 2.0 

Again we find, notwithstanding the decidedly su- 
perior acquisition of the material studied (see Table 



Average Mark for New 




Material. 


Positive 


Marks after 


marlis. 


deduction. 


16.3 


15.6 


3.5 


4.0 


15.7 


14.9 


5.8 


5.2 



98 INDUCTIVE VS. DEDUCTIVE METHODS. 

XX) on the part of the deductive group, that they are 
inferior to the other in their power to attack new 
material of an analogous nature. Four boys in the 
deductive group completely failed to make a reason- 
able application of their old knowledge, obtaining 
only 6, 7, 2 and 4 marks, respectively, whilst only one 
boy in the inductive group failed to do so, and he 
obtained 8 marks. 

Let us now see, as we have in previous cases, how 
far this difference is to be found for the weaker as 
well as for the abler children of each group : 

Table XXIII, shoiciiig the irork of the tico groups compared, sec- 
tion by section, in the Prcliminarii Tests and in the Test of 
Application to Neiv Matrrial (positive marks, and positive 
inarks after the deduction of the negative marks). 

Group Inductively Taught. 

Marks for Application to 

Marks New Material. 

in four No. , -> 

preliminary of Positive After 

tests. boys. only. deduction. 

70 and over 2 18.5 18.0 

60 to 70 4 18.3 17.0 

50 to 60 5 17.8 17.6 

40 to 50 8 15.5 15.0 

Below 40 6 14.0 13.2 

Group Deductively Taught, 

Marks for Application to 

Marks New Material. 

in four No. r ^ 

preliminary of Positive After 

tests. boys. only. deduction. 

70 and over 2 18.0 10.5 

60 to 70 4 15.0 14.2 

50 to 00 17.8 17.2 

40 to 50 7 14.0 13.3 

Below 40 6 15.3 14.6 

Except in the case of the least proficient section of 
boys at the bottom of each group, there seems to be 



THIRD SERIES OF EXPERIMENTS. 99 

an advantage all along the line in favor of the in- 
ductive group. When, therefore, the tests are tests 
of the application of knowledge rather than an exact 
reproduction of it, we are, perhaps, entitled, on the 
whole, to conclude that inductive methods are the 
better. It may be noted that whilst the marks for the 
inductive group proceed regularly downwards from 
the highest section to the lowest, those for the corre- 
sponding sections of the deductive group do not. The 
variability for this group is disproportionally high, 
due, doubtless, to the psychological fact that for 
some children of this age the step from knowledge 
to the application of it is a very considerable one; 
whereas, of course, the children of the other group 
had been through a process of applicable method 
when they had received their inductive lesson. The 
variability of the work is, however, decidedly high, 
and the difference between the means of the work of 
the two groups is very small; and did this experi- 
ment stand alone, I should hesitate before putting 
much confidence in the conclusion which I have indi- 
cated above. But its consilience with the previous 
results lends strength to the conclusion, especially 
when the differences in the conditions under which 
it was obtained are taken into account. 



VII. FOURTH SERIES OF EXPERIMENTS. 

1. General Plan. 

In the experiment now to be described, just as in 
those previously recounted, the work was done with 
all the children of one class, under one teacher, with 
the same curriculum of study, and working according 
to the same time-table of instruction. The experi- 
ment was carried out in a municipal higher grade 
school for boys, an elementary school situated in a 
somewhat mixed neighborhood. The class chosen 
for the experiment was the First Class in the school, 
containing 35 boys, graded as Ex. VII on the English 
standard system of school grading, of an average age 
of approximately 131/2 years.* 

The teacher of the class had a theoretical acquaint- 
ance with psychological work, and had already car- 
ried out some observations in educational psychol- 
ogy. He was, especially, capable of temporary dis- 
sociation between the pedagogic and psychologic 
attitudes — a necessary capacity in an experimenter. 
Beyond this, he was a first-rate teacher who varied 
his methods according to the subject-matter with 
which he was dealing. 

As in previous cases, the class was divided into 
two equal groups on the results of tests in spontane- 

*In America this would constitute Grade VIII, or rather, per- 
haps, the First Year of High School. 

100 



FOURTH SERIES OF EXPERIMENTS. 101 

ous definition, but the test on which the division was 
effected was not the same as that used for the pur- 
pose in the former tests. But, as before, there were 
tests of immediate and tests of deferred reproduc- 
tion, and a test of application to new material of an 
analogous kind. Further relevant conditions will 
be given in the details which follow. 



>s' 



2. The Preliminary Tests and the Method of 
Marking. 

The first test in this series was the spontaneous 
definition of squares, triangles, oblongs, and diam- 
eters of circles, which were drawn in the way already 
indicated, and the questions (with which by now the 
reader will be quite familiar) : ^'What is a square?" 
etc., were set for written answers. The papers were 
marked on the system of units which has already 
been described, and an average mark was gained of 
19.1 out of a possible maximum of 30. This, as might 
have been expected, was by far the highest mark that 
had been obtained by any class doing this test. It 
was not proposed to use this test on squares, tri- 
angles, etc., for the purpose of dividing the class, but 
it served a useful purpose as a preparatory exercise. 

Next week the teacher taught all the children of 
the class how to arrive at the definitions inductively 
in the way that I have described in the first series of 
experiments (p. 33). This lesson also rendered val- 
uable service. It gave full opportunity to all to un- 
derstand quite clearly what they had to do when they 
were set to attack the preliminary test on which the 
class ivas to be divided. 

The questions used for the preliminary test were 



102 INDUCTIVE VS. DEDUCTIVE METHODS. 

the same as those which, in the previous schools, had 
been used as a test of the power of application to new 
analogous material. In one sense it is, of course, in 
this case, also a test of application to new material, 
for one inductive lesson on the square, etc., had 
already been given. We may, indeed, look upon our 
division into two ' equal groups ' in this case as being 
effected during the course of a series of lessons in- 
stead of at the very beginning of it. 

The questions were: ''What is a rhombus?" 
"What is a trapezium?" "What is a rhomboid?" 
and "What is a diagonal of a square?" The an- 
swers were marked on the system of units which has 
already been described.* 

One or two of the papers worked may be of service 
in enabling an experienced teacher to gauge the men- 
tal level of the boys taking these tests. 

Edward S , aged 14 years 8 months, wrote : 

1. A rhombus is a figure, it has 4 equal straight lines, it has 
angles, there are 4, 2 equal large ones and 2 equal small ones. 

2. A trapezium is a figure, it has 4 straight lines of different 
lengths, it has angles, there are 4, all of different sizes, any shape. 

3. A rhomboid is a figure, it is enclosed by 4 straight lines, 2 
equal long ones, two equal short ones. It has 4 angles, 2 equal 
small ones and two equal large ones. 

4. A diagonal is a straight line going from one corner to the 
other of a square terminating at both ends dividing the square into 
2 triangles. 

This is not the best paper ; there are four boys in 
the class who get higher marks, but it is obvious that 
we are here dealing with a very different mental 
level, geometrically, from those at which we have 
previously worked. With the table of units at hand, 



*The reader is recommended to turn to page 41 for the list of 
correct units of description. 



FOURTH SERIES OF EXPERIMENTS. 103 

it is quite easy to mark this paper. The only diffi- 
culty occurs in the case of the last definition, in which 
the qualification opposite is omitted when the corners 
of the square are mentioned. It is held, however, 
that the statement '^ dividing the square into 2 tri- 
angles" is equivalent to the limitation of from one 
corner to the 'opposite' corner. A total of 37 posi- 
tive marks was gained — 11 for the definition of 
rhombus, 8 for the definition of trapezium, 14 for the 
definition of rhomboid, and 4 for the definition of a 
diagonal of a square. There are no 'bad errors.' 
Charles B , aged 13 years 9 months, wrote : 

1. A rhombus is a figure or drawing consisting of 4 straight 
lines. All the lines are the same length and the two lines opposite 
one another are parrallel to one another. It has 4 corners and four 
equal angles. 

2. A trapezium is a drawing. It has 4 straight lines, 4 angles, 
all the lines are of different lengths. It has two long and two short 
sides. The angles are all different. 

3. A rhomboid is a drawing consisting of 4 lines which are 
straight. It has 2 long and 2 short. The 2 short are parrallel to 
one another, and the 2 long are parrallel to one another. It has 
4 corners and four angles. The 2 long sides are the same length 
and the 2 short are the same length. All the angles are not equal. 

4. A diagonal is a line, must be straight. It is drawni from one 
corner to the one opposite. It passes through the centre of the 
figure. It does not go outside the figure. It must touch the 
corners. 

This also is a good paper, gaining one mark more 
than the average for the whole class. The marks are 
quite easily given. The definition of rhombus gains 
9 positive marks. There is one 'bad error' — the four 
angles are not equal; but throughout this experiment 
we worked with positive marks only. The definition 
of trapezium gains all the positive marks possible 
on our scale of marking, namely, 8. It is interesting 
to note that the term 'corners' is not in C. B.'s mind 
synonymous with ' angles. ' In every one of his defi- 



104 INDUCTIVE VS. DEDUCTIVE METHODS. 

nitions in which the confusion can occur lie makes 
the same duplication, but these duplications are not 
'bad errors' according to our system of marking. 
The definition of rhomboid receives 12 positive 
marks, and the definition of a diagonal of a square 
receives full marks, namely, 4. The total marks for 
this paper amount to 33. It is scarcely necessary to 
multiply examples; sufficient have been given to 
show how much more competent, geometrically 
speaking, these boys are than those with whom we 
worked in the previous boys' school. I turn now to 
the chronology of the whole of the experiment. 

3. Chronology of the Experiment. 

The first test in this series was given on Friday 
at 9.30 A. M., September 29, immediately after Scrip- 
ture lesson. In this test the boys were asked, un- 
taught and unaided, to define square, triangle, etc. 
Exactly one week later all the boys in the class were 
taught inductively how to arrive at the definitions 
of square, triangle, etc. On Tuesday, October 17, 
at 11 o'clock, immediately after recreation, the test 
used as a Preliminary Test in this school, ' ' What is 
a rhombus 1 ' ' etc., was given, on the results of which 
the class was divided into two equal groups. In this 
experiment one test only was given for purposes of 
division. It was hoped that the preparatory work 
with the squares, triangles, etc., together with the 
greater age and proficiency of the children, would 
result in the necessary steadiness, and that the boys' 
variability would be so small that one test would 
suffice. 

On Thursday afternoon, October 19, from 2.30 to 
2.50, one of the groups was taught inductively how to 



FOURTH SERIES OF EXPERIMENTS. 105 

arrive at the definitions of rhombus, trapezium, 
rhomboid, and diagonal of a square. Exactly the 
same method was followed as that used by me in the 
first and second experiments. The teacher of the 
class had heard me 'teach' the definitions, so there 
was no danger that he would vary the method essen- 
tially ; but a minor variant was employed. He jotted 
down on the blackboard (which I did not), in an ab- 
breviated form, the 'units' of description as the boys 
supplied them. His argument for the variation was 
that the boys who were going to study the definitions 
deductively would have visual memories of verbal 
descriptions to help them, and that the inductively 
taught group ought also to have some visual verbal 
memories to assist them. Whilst the boys of one 
group — Group A — were being taught the definitions, 
the other group — Group B — went into the school hall 
and had a reading lesson under a student-teacher. 
From 2.55 to 3.15 the boys of Group A went into the 
school hall and took the reading lesson, whilst Group 
B came back to their own teacher and studied the 
definitions of rhombus, etc., which had been con- 
structed from the spontaneous descriptions of the 
Preliminary Test and had been already written in 
preparation upon a blackboard, with the appropriate 
drawings. 

Definitions of Rhombus, Trapezium, Rhomboid and 

Diagonal of a Square in the Form in Which 

They Were Given to the 'Deductive' 

Group to Study. 

A rhombus is a figure enclosed by 4 equal straight 
lines. Two sides opposite are parallel, and the other 
two sides opposite are parallel. It has 4 angles, 2 



106 INDUCTIVE VS. DEDUCTIVE METHODS. 

large and 2 small. The 2 large angles are equal and 
opposite, and the 2 small angles are also equal and 
oj^posite. 

A trapezium is a figure enclosed by 4 unequal 
straight lines. It contains 4 unequal angles. 

A rliomhoid is a figure enclosed by 4 straight linos, 
2 long and 2 short. Two long sides are equal, oppo- 
site and parallel, and the two short sides are equal, 
opposite and parallel. It has 4 angles, 2 large and 2 
small. The 2 large angles are equal and opposite, 
and the 2 small angles are equal and opposite. 

A diagonal of a square is a straight line which 
starts at an angle and passes across the square to the 
opposite angle. 

The boys were told to study the definitions, and 
they, as well as the boys of the inductive group, were 
made aware that they would be required to answer 
questions on them. The time from 3.15 to 3.30 was 
spent by the boys of both groups in the playground. 
At 3.30 all the boys returned to their classroom ; the 
questions, "What is a rhombus?" etc., which had 
been written on the blackboard, were exposed to 
view, and the boys wrote the answers. There was 
one other variant from the method which I had used 
myself, for the drawings of the figures were placed 
before the boys whilst they were answering the ques- 
tions in their tests of reproduction. 

Exactly one week later, on Thursday, October 26, 
at 3.30 P. M., the boys of both groups worked a test in 
deferred reproduction, following immediately upon 
the recreation interval, as in the test of immediate 
reproduction. 

Two weeks after this test, at 10 o'clock in the 
morning, on Wednesday, November 8, following two 



FOUKTH SERIES OF EXPERIMENTS. 107 

short lessons on Scripture and French reading, the 
test of Application to New Material was given. 

Perhaps a summarized note showing the main 
chronological issues involving differences from other 
experiments may be of service. First, both groups 
had inductive teaching, as well as inductive practice, 
before the Preliminary Test. There was one pre- 
liminary test, and only one. The test of deferred re- 
production was given one week after the test of im- 
mediate reproduction. The test of application to 
new material was given three weeks after the teach- 
ing and learning which we were relying on to differ- 
entiate the groups, and two weeks after the test of 
deferred reproduction. 

4. The Tests of Immediate and Deferred Repro- 
duction. 

In these tests all the boys in the class answered in 
writing the following questions: ''What is a rhom- 
bus?" etc. The questions were written on the black- 
board, and the drawings of the rhombus and other 
figures were again shown to the boys. I have already 
pointed out that this was a variation on the method 
previously adopted. 

5. The Test of Application to Neiv Material. 

Drawings of hexagons, pentagons, tangents and 
quadrilaterals (similar figures), with the names ap- 
pended, were shown to the boys thus : 



108 INDUCTIVE VS. DEDUCTIVE METHODS. 

HEXAGONS. 





PENTAGONS. 






TANGENTS TO CIRCLES. 
(The tangents are drawn in dots.) 





The sides of LMNO were each IV2 times the corresponding sides 
of ABCD, so that no easily recognizable ratio should appear. The 
figures were drawn so that CD and NO were not quite in the same 
straight line. 

In the diagrams actually used the tangents were continuous Imes 
drawn in red. 



FOURTH SERIES OF EXPERIMENTS. 109 

Then the following questions were written on a 
blackboard and the boys required to answer them in 
writing : 

1. ''What is a hexagonl" 

2. "What is a pentagon!" 

3. "What is a tangent to a cicle?" 

4. "In how manv ways is ABCD like 

LMNO!" 

The boys were allowed, nay encouraged, to give 
thought and time to their answers. It will, doubtless, 
be remembered that no time limits were imposed in 
any of the tests and exercises in these experiments. 
The system of marking the papers could, no doubt, 
be inferred by analogy from the units of correct de- 
scription which the boys and girls have given in other 
cases and which we have adopted. But it is unneces- 
sary for us to infer what our units ought to be; they 
emerge quite clearly from a consideration of the pa- 
pers actually worked. 

Let me give one or two by way of illustration be- 
fore listing the units on which the boys' papers were 
marked. 

Frederic R , aged 13 years 11 months, who 

worked in the deductive group, wrote : 

1. A hexagon is a figure enclosed by six equal straight lines. It 
has six angles, all equal. The two opposite sides are parallel in 
the three cases. 

2. A pentagon is a figure enclosed by five equal straight lines. 
It has five angles all equal. None of the sides are parallel to each 
other. 

3. A tangent to a circle is a straight line any length, which must 
touch the side of the circle anywhere, but must not cut it. 

4. The first thing why ABCD differs from LMNO is its size. 
The 4 angles are the same in both figures. The 4 straight lines are 
the same only in proportion, LMNO is about half the size again 
as ABCD, M.N.O. angles are the same as B.D.C. only the sides are 



110 INDUCTIVE VS. DEDUCTIVE METHODS. 

different lengths. A. angle is exactly the same as L. angle. Both 
the figures are exactly the same shape. The only thing why one 
is different from the other is in size. 

Even without a list of units of correct description 
it is not difficult to assess this paper. The definition 
of hexagon receives a mark for 'figure,' four marks 
for ''six equal straight lines," three marks for "six 
angles equal," and six marks for noting that there 
were three pairs of opposite sides, and that three 
pairs were parallel. F. R. thus obtains a total of 14 
marks for his definition of hexagon. The definition 
of pentagon receives eight marks — one for 'figure,' 
four for "five equal straight lines," and three for 
"five angles equal." The definition of tangent re- 
ceives three marks — one for 'line,' one for 'straight,' 
and one for "touching the side of the circle." A 
boy's conception of touching would be satisfied if the 
line impinged upon the circumference of the circle 
in such a way that, if produced, it would cut the cir- 
cumference. Consequently it is necessary to add the 
limitation 'if produced, will not cut the circle.' The 
fourth answer is a good one, but it is unfortunate that 
the boy is bothered by the notion that he has to find 
differences, which every now and again intervene 
among the similarities. He calls ABCT) and LMNO 
both 'figures,' for which he receives a mark; for '4 
angles' he receives two more ; for noting that the four 
angles are equal, each to each, in the two figures, he 
receives four marks; for "4 straight sides" three 
more, and for tlie similar proportionality of the sides 
he obtains four more. Finally, he notes that the fig- 
ures are alike in shape, for which he receives a mark. 
F. R.'s total mark for this answer is 15, and his mark 
for his whole paper 40. His marks were 38 for his 



FOUETH SERIES OF EXPERIMENTS. Ill 

preliminary test, 49 for immediate reproduction 
after teaching, 49 for deferred reproduction, and 40 
— the present mark — for application to new mate- 
rial. If these are compared with the average marks 
given later, it will be seen that he is five or six marks 
ahead of the average throughout the entire series. 
I will give one or more worked papers before setting 
out the units of correct definition which were ac- 
cepted as the basis of marking. 

Robert S , aged 14 years, who worked in the 

inductive group, wrote : 

1. A hexagon is a figure enclosed by six, straight, equal, sides. 
The opposite sides are equal and parallel. One side is exactly 
balanced by the opposite one. It has six angles, which are all 
equal. Three are on one side and three on the other. 

2. A pentagon is a figure enclosed by five straight sides. They 
may be equal or unequal. No sides are opposite and no sides are 
parallel. It have five angles. They may be equal or unequal. 

3. A tangent to a circle is a straight line. It may be drawn at 
any angle. It must touch the circle but not cut it. 

4. Both have five sides. The base in each case is horizontal. 
They have five angles each. The angles are the same number of 
degress in each case. There are two large ones and two small ones. 
The two large ones are formed by the base and sides and the two 
small ones from the top and sides. The smallest angle is A in 1 
and correspondonds with L in 2. The largest C in 1 and N in 2. 

The definition of hexagon obtains 14 marks — one 
for ' figure, ' four for ' ' six straight equal sides ; " six 
for noting that there are three pairs of opposite 
sides, and that they are parallel, each to each, and 
three marks for ''six equal angles." The definition 
of pentagon obtains six positive marks — one for "fig- 
ure," three for "five straight sides," and two for 
"five angles." The statement 'no sides are opposite 



112 INDUCTIVE VS. DEDUCTIVE METHODS. 

and no sides are parallel' is held to be of too negative 
a nature for inclusion within the definition. To say 
that the sides and the angles may be equal or unequal 
would be accounted 'bad errors,' though, as I have 
said before, we did not tabulate the 'bad errors' in 
this fourth experiment. The definition of tangent 
receives three marks — one for 'straight,' one for 
'line,' and one for 'touch the circle.' "It may be 
drawn at any angle" is too vague to be regarded as 
either positive or negative. The point is missed that 
the tangent, if produced^ will not cut the circle. In 
the fourth answer there are two curious errors. Tlie 
figures have 4 sides and 4 angles, and not 5, as R. S. 
says. He obtains marks for mentioning 'sides' and 
'angles' as pertaining to both. Nearly all the rest 
of the answer is occupied with the equality of the 
angles each to each, for which 4 marks are obtained. 
One mark is gained for noting that the base lines in 
each case are horizontal ; that is regarded as equiva- 
lent to noting that their inclination is the same. This 
marking yields a total of 30 positive marks, with 4 
'bad errors.' I give this paper because I wish to 
make it quite clear that boys inductively taught 
could quite well make 'howlers' as well as boys de- 
ductively taught, though these boys, in both groups, 
make extremely few. R. S.'s other marks were 26, 
42 and 43 ; in all cases, except that of the Deferred 
Reproduction Test, well below the average. Prob- 
ably the perusal of the papers given above may make 
clearer the usefulness of the units of correct defini- 
tion which are now appended. 



FOUETH SERIES OF EXPERIMENTS. 113 

Units of Correct Description or Definition of 
Hexagon, etc. 

1. A hexagon is a figure. 
It has sides or lines. 

It has straight sides or Hnes. 
It has equal sides. 
It has six equal sides. 
Two sides are opposite. 
Two other sides are opposite. 
And the two other sides are opposite. 
Two opposite sides are parallel. 
Other two opposite sides are parallel. 
And the other two opposite sides are parallel. 
It has angles. 

Its angles are six in number. 
And they are equal. 
Its angles are greater than right angles. 
(A total of 15 points.) 

2. A pentagon is a figure. 
It has sides or lines. 
Its sides are straight. 
The sides are equal. 
There are five sides. 

It has angles. 

Its angles are five in number. 
The angles are equal. 
And they are greater than right angles. 
(A total of 9 points.) 

3. A. tangent to a circle is a line. 
It is a straight line. 

The line touches the circle. 
And, if produced, does not cut it. 
(A total of 4 points.) 



114 INDUCTIVE VS. DEDUCTIVE METHODS. 

4. ABCD and LMNO are both figures. 
They both have sides. 
They both have straight sides. 
Their sides are in both cases unequal. 
And they are 4 in number in both figures. 
They both have angles. 
Their angles are 4 in number. 
And are in both cases unequal angles. 
BA is the same fraction of LM. 
As BC is of MN. 
As CD is of NO. 
As AD is of LO. 

BA has the same slant or is parallel to LM. 
BC has the same slant or is parallel to MN. 
CD is parallel to NO. 
And AD is parallel to LO. 
The angle at A equals the angle at L. 
The angle at B equals the angle at M. 
The angle at C equals the angle at N. 
The angle at D equals the angle at 0. 
The figures have the same shape. 
(A total of 21 points.) 

It is, of course, not urged that the common proper- 
ties of the figures have been exhaustively enumer- 
ated, but only that the units of correct description 
are such as are actually used by boys and are service- 
able for marking papers in such experiments as 
these. 

6. Results. 

First, let me give the coefficients of correlation be- 
tween the results for the various exercises in so far 
as they may be of value. The marks for the Prelimi- 



FOURTH SERIES OF EXPERIMENTS. 115 

nary Test in the A Group were very closely corre- 
lated with those of the B Group ; the boys were most 
successfully paired in the two groups, from the best 
downwards to the worst. Worked out on the regular 
formula, the coefficient of correlation amounted to 
+ .98. The results of the test in immediate repro- 
duction correlated with that of deferred reproduction 
to the extent of -f .752 in the inductive group and 
-f .777 in the deductive group. There was a falling 
off on the average of about one unit in the marks. 
There were 7 cases out of 34 in which the mark for 
deferred rej)roduction was higher than for immediate 
reproduction, 10 cases in which it was the same, and 
17 cases in which there was a decline. The decline 
of the whole class was from an average mark per boy 
of 44.94 to 43.50, with mean variations approximat- 
ing to 4 in both cases, and a correlation coefficient 
between the results of immediate and deferred re- 
production of 4- .78. Though the difference is small, 
we are entitled statistically to say that there is a 
general tendency to decline, since the difference be- 
tween the means is from three to four times the 
probable error of that difference. A general slight 
decline seems, therefore, clear. The inductive group 
falls from 45.2 to 43.5; the deductive from 44.7 to 
43.5. But the fall is too irregular to enable us to con- 
clude that there is any greater tendency to loss on 
the part of the children inductively taught than of 
those deductively taught. 

Let us now consider the results of the test on new 
material. It is clear that the difference between the 
results of the two groups is very small in this school, 
though it favors the inductive group, as, indeed, is 
the case in all the experiments. But the variability 



116 



INDUCTIVE VS. DEDUCTIVE METHODS. 



is such that without very high positive correlation 
between the two series the probable error of the dif- 
ference between the means will be considerable. 

Now let me give the average results in gross, treat- 
ing the groups as wholes. There were 17 boys in 
each group : 

Table XXIV, shoioing the tvork of the Inductive and Deductive 
groups compared, in the Preliminary Test, in the Tests of Im- 
mediate and Deferred Reproduction, and in the Test of Appli- 
cation to Netv Material. 







Test of 


Test of 






Pre- 


immediate 


deferred 


Test on 




liminary 


repro- 


repro- 


new 


Inductive Group : 


test. 


duction. 


duction. 


material, 


Average mark . . 


. 32.06 


45.18 


43.53 


35.65 


M. V 


. 3.13 


3.67 


3.97 


3.24 


Deductive Group: 










Average mark . . 


. 32.12 


44.71 


43.47 


35.00 


M. V 


, . 2.98 


4.69 


4.80 


3.53 



The boys of the Inductive Group appear to hold 
the advantage throughout, though they were slightly 
weaker in the Preliminary Test. A closer analysis 
of the results is given in the next table : 

TaWe XXV, shoxcing the work of the Inductive and Deductive 
Oroups compared, section by section, in the Preliminary Test, 
the Tests of Immediate and- Deferred Reproduction, and the 
Test of Application to New Material. 



Inductive Group. 



Marks in No. 

preliminary of 

test. boys. 

Over 35 4 

30 to 35 6 

25 to 30 7 



Marks in No. 

preliminary of 

test. boys. 

Over 35 4 

30 to 35 6 

25 to 30 7 



Pre- 


Immediate 


Deferred 


> 


liminary 


repro- 


repro- 


New 


test. 


duction. 


duction. 


material. 


37.50 


47.00 


45.50 


37.50 


32.50 


47.00 


45.83 


37.66 


28.57 
Deductiv 


42.57 
e Group. 


40.42 


32.85 


1 


-Average Mark per Boy. , 


Pre- 


Immediate 


Deferred 




liminary 


repro- 


repro- 


New 


test. 


duction. 


duction. 


material. 


37.50 


47.25 


47.75 


37.00 


32.33 


44.16 


43.16 


36.60 


28.85 


43.71 


41.28 


32.42 



FOURTH SERIES OF EXPERIMENTS. 117 

Only in one test — that of application to new mate- 
rial — does there appear to be a regular sectional ad- 
vantage on the side of the inductive group, both for 
the weaker as well as for the stronger boys. In both 
reproductive tests the balance of advantage shifts 
from side to side. 

We may justifiably conclude that the results of this 
experiment, having regard to the greater age and 
mental ability of the children, are consilient with 
those of the former researches. The inductive 
method has shown itself the better when application 
to new analogous material is the test employed. We 
are unable to say with any confidence which of the 
two groups has been the more successful in immedi- 
ate and deferred reproduction. The average results 
are slightly in favor of the inductive group, but the 
balance of advantage fluctuates from side to side, and 
is decidedly uncertain. But this is the fourth case 
in which the inductive method has shown itself supe- 
rior in application to new material, and the second 
case in which the inductive method has equaled the 
other, even for purjDOses of reproduction. In both 
these classes there had been much previous inductive 
teaching. But it must be remembered as well that 
the class of much younger hoys, in which the deduct- 
ive group scored heavily in reproductive work, were 
also accustomed to much inductive work. Age ap- 
pears to be a factor ; perhaps it is the younger chil- 
dren who reproduce better on a deductive and memo- 
riter method. This hypothesis will be put to the test 
in the last of this series of experiments. 



VIII. FIFTH SERIES OF EXPERIMENTS. 

1. General Plan. 

Just as in the previous experiment, a whole class, 
working under one teacher, and according to the 
same syllabus of instruction, with the same time- 
table of work, was divided into two equal groups on 
the results of a test in the definition of geometrical 
forms, which the boys attempted, untaught and un- 
aided. Then one group worked inductively and the 
other deductively. Tests were given immediately 
after the teaching and learning, also in deferred re- 
production a week later, and in reproduction, still 
further deferred, about seven weeks after the first 
test of deferred reproduction. About two weeks 
after the teaching and learning a test of application 
to new material was given. The boys who did the 
work were graded as Standards VI, a, and VII ; their 
average age was 12 years Qi'o months, and their 
teaching generally had been clear and efficient, but 
had tended rather towards deductive than inductive 
methods. The school was situated in a poor neigh- 
borhood in the southeast of London, and the average 
mental ability of its pupils was low ;* but the boys of 
the highest class, with whom the experiment was 
made, were by no means without ability ; in fact, in 

♦The natural mental ability of the pupils of this school was 
well known to me from the results of a number of mental tests 
which had been applied to every child over eight years of age. 

119 



120 INDUCTIVE VS. DEDUCTIVE METHODS. 

consequence of certain exigencies of organization, 
the class contained more children of ability than 
would ordinarily be found in a top class of such a 
size in a school of this social type. 

2. The Preliminary Tests and the Method of 
Marking. 

The Preliminary Test, on the results of which the 
boys were divided into two equal groups, was the 
same as that used in the experiment just described, 
which took place in the higher grade school. The 
teacher had already used the questions: "What is a 
square ? " " What is a triangle ? " " What is an ob- 
long?" and "What is a diameter of a cicleT' (with 
the consideration of the appropriate drawings) as a 
kind of general propaedeutic to the experimental 
series, and the boys had already been shown in- 
ductively how to work out the definitions of square, 
triangle, oblong and diameter just as they had in the 
higher grade school. 

The Preliminary Test, given two or three weeks 
later, consisted in the questions: "What is a rhom- 
bus?" "What is a trapezium?" "What is a rhom- 
boid?" and "What is a diagonal of a square?" The 
appropriate drawings were shown and the boys at- 
tempted to answer the questions. I give below one 
or two of their worked papers. 

William L , aged 13 years 8 months, wrote : 

1. A Rhombus is a figure with all side equal two sides slope at 
60° and the other two run parallel. 

2. A Trapezium is a figure with four unequal sides, and it as a 
right angle in it. 

3. A Rhomboid is a figure with two long sides equal and two 
short sides equal, but none of the corners have right angles. 

4. A Diagonal of a Square is the distance across from corner to 
another corner which slopes at 45°. 



FIFTH SEEIES OF EXPERIMENTS. 121 

W. L., in his first definition, gains a total of six 
marks. His second definition receives four marks. 
"It as a right angle in it" is not true as applied to 
all the trapeziums; it is a 'bad error,' but there are 
so few of these that they are not tabulated. The 
definition of rhomboid receives eight marks. The 
negative statement that there are no right angles, 
though correct, receives no mark, as we could hardly 
have made allowance for all the negative statements 
which may be truly made about the figures. The defi- 
nition of diagonal receives two marks only — one for 
'distance' and one for "from corner to another 
corner." W. L.'s paper receives a total of 20 posi- 
tive marks. It is one of the best papers worked in 
the class, and is assessed considerably above the av- 
erage mark, which is 12.25 for this preliminary test. 

Let me now give the paper of a boy who is among 
those toward the bottom of the lists. 

Frank B , aged 12 years 4 months, wrote : 

1. A Rhombus is a square turned in shape. 

2. A Trapezium is a figure all sides unequal. 

3. A Rhomboid is an oblong with the two smallest perpendicular 
lines slanting. 

4. Diagonal of a square is a line drawn from top to bottom of 
the corners. 

As we have seen in former cases of 'unintelligent' 
children, the similarities between these figures and 
those which they have previously dealt with are ap- 
prehended, even to the extent of error, for a rhombus 
is not a square. That a square is one shape and a 
rhombus is another shape is probably dimly under- 
stood by the boy ; he is giving, perhaps, what he con- 
ceives to be a genetic definition of a rhombus, but he 
receives no marks for it on our system of marking. 



122 INDUCTIVE VS. DEDUCTIVE METHODS. 

For his definition of trapezium he obtains three 
marks. The rhomboid he defines genetically; his 
definition is worth, perhaps, two marks — one for 
'lines' and one for ''two smallest lines." His defini- 
tion of diagonal is worth two marks ; he describes it 
as a 'line' and notes that it goes from one corner to 
another. F. B. thus receives a total of seven marks, 
which is a little above half the average mark for the 
class. 

The two examples of worked papers, given above, 
will enable teachers to see the limits of the mental 
level with which we are dealing. These boys are very 
obviously much below the first-class boys of the 
higher grade school whose work we considered in 
the experiment previously described. 

3. Chronology of the Experiment. 

First of all came the inductive work with the 
squares, triangles, oblongs and diameters of circles. 
This was done with all the class. A week or so later, 
on Wednesday, October 11, at 10 A. M., follomng 
immediately upon Scripture lesson, the Preliminary 
Test was given, on the results of which the boys were 
divided into two equal groups. Most of the boys had 
finished their work in 20 minutes, though no one was 
hurried, and one or two took a few minutes longer. 
On Thursday, October 12, immediately after registra- 
tion, the teacher of the class taught one of the groups 
how to arrive inductively at the definitions of the 
geometrical figures which thej^ had attempted in the 
Preliminary Test. The teacher had heard me teach 
similar definitions and was well acquainted with the 
method as I employed it. The teaching took 22 min- 



FIFTH SERIES OF EXPERIMENTS. 123 

utes, from 2.14 to 2.36 P. M. Wliilst the one group 
was being taught by their own teacher, the other 
group, under another master, studied the definitions 
with reference to the drawings of the figures which 
were appended. They knew that the exact words of 
the definitions were not to be required, but that they 
might use them if they chose. Both groups of boys 
were aware that they were to be tested immediately 
afterwards on what they had been taught or learnt. 
The definitions given to the 'deductive' group ran as 
follows : 

Definitions of Rhombus, Trapezium, Rhomboid and 

Diagonal of Square to Which Appropriate 

Drawings Were Appended* 

1. A Rhombust is a figure with four straight 
equal sides; the opposite sides are parallel. It has 
four corners, two big ones opposite and equal, and 
two smaller ones opposite and equal. 

2. A Trapezium is a figure or shape with four 
straight unequal sides and four unequal corners. 

3. A Rhomboid is a figTire with four straight 
sides. The two long sides are opposite, equal and 
parallel. The two short sides are opposite, equal 
and parallel. It has four corners, two big and two 
small. The two big ones are equal and opposite, and 
the two small ones are equal and opposite. 



*The drawings may be seen on page 39. 

fPerhaps a slight amendment miglit usefully have been made in 
this definition of the rhombus ; it is not clear on this wording that 
there are tico pairs of opposite sides which are parallel ; the form 
of words used in the previous experiment seems more satisfactory. 



124 INDUCTIVE VS. DEDUCTIVE METHODS. 

4. A Diagonal of a Square is a straight line 
drawn from one corner to the opposite corner. 

At 2.40 P. M., a few minutes after the teaching and 
learning, both groups answered in writing the ques- 
tions: "What is a rhombus!" etc. No time restric- 
tions were laid down ; each boy was permitted to go 
on until he could do no more, but the superiority of 
the pace of the boys who had learnt the definitions 
was evident in this and in all succeeding exercises. 
After a lapse of one week, at the same hour in the 
afternoon, namely, 2.40, and on the same day of the 
week, Thursday, October 19, both groups answered 
the questions : ' ' What is a rhombus I ' ' etc. This will 
be referred to as the first test of deferred reproduc- 
tion. None of the boys were aware that they were 
ever again to be required to answer these questions ; 
it was only the test of immediate reproduction of 
which they had been forewarned. 

One week later, again on Thursday at 2.40 P. M. 
(October 26), the boys worked a further test — a test 
of application to new material — and on Thursday, 
December 7, at 2.40 P. M., two months after the test 
of immediate reproduction, a second test of deferred 
reproduction was given, in which the old questions, 
"What is a rhombus?" etc., were repeated; and, as 
before, the boys answered them in writing. 

4. The Tests of Reproduction. 

These were in all cases the same. They consisted, 
as previously stated, of answers in writing to the 
questions : ' ' What is a rhombus ? ' ' etc. One or two 
papers to indicate what these boys could do after 
teaching and learning may be of interest. 



FIFTH SERIES OF EXPERIMENTS. 125 

Thomas G , aged 13 years 6 months, one of the 

best boys who worked in the deductive group, in his 
exercise in immediate reproduction, wrote : 

1. A Rhombus is a figure with four equal straight liues. The 
opposite lines are parallel. It has four corners, one pair of oppo- 
site corners being equal and the other pair of opposite corners being 
equal. 

2. A Trapezium is a figure with four unequal sides, and four 
unequal corners. 

3. A Khomlwid is a figure with four straight sides, two long 
sidesi and two short ones. The two long ones are equal and opf>o- 
site each other, and the two short ones are equal and opposite 
each other. The figure has four corners, two big ones and two 
little ones, The two big ones are equal and opposite, and the two 
little ones are equal and opposite. 

4. A diagonal of a square is a straight line drawn from one 
corner to the opposite corner. 

This is an excellent pajjer ; the definition of rhom- 
boid, for example, where it differs from the wording 
of the definition studied, is better than the definition 
we provided. The boy rightly says ''two long and 
two short" sides, before speaking of "The two long" 
sides. Our own definition is faulty in that respect. 
The word 'The' is not only distinguishing, but rela- 
tive, and, indeed, very often distinguishing because 
relative. Let us mark the paper in accordance with 
the list of units of correct description given on 
page — : 

The definition of rhombus receives single marks 
for 'figure,' 'four,' 'equal,' 'straight,' 'lines,' 'four,' 
'corners,' and eight marks for noting two pairs of 
opposite, parallel sides and two pairs of opposite 
equal corners — a total of 15 marks. For his defini- 
tion of trapezium he receives obviously every mark 
but one. He has omitted 'straight' in his descrip- 
tion of the sides, thus receiving seven marks out of 
eight. Every possible point on our system of mark- 



126 INDUCTIVE VS. DEDUCTIVE METHODS. 

ing is scored by his definition of rhomboid, with the 
exception of two ; he omits the two pairs of parallels, 
thus receiving 18 marks. The definition of a diag- 
onal of a square receives full marks, namely, four. 
F. G. 's total mark for his immediate reproduction is 
44, which is much above the average of his group. 

In Ms next test, one week later, he loses four marks 
on his first definition, for he now omits to note the 
two pairs of opposite parallel sides. His mark for 
trapezium remains unchanged. In his definition of 
rhomboid he now notes the two pairs of parallels, 
which he omitted to do in his test of immediate re- 
production, and on this occasion receives full marks, 
namely, 20. The definition of a diagonal of a square 
remains unchanged. F. G., therefore, has gone down 
two marks in one week. Let us see how many he has 
lost seven weeks after this. The definition of rhom- 
bus suffers most; the parallelism of the opposite 
sides does not reappear, and it is doubtful whether 
F. G. remembers that there are tivo pairs of opposite 
equal angles, for his expression is dubious. He has 
now lost four of the marks he originally obtained for 
this definition. The definition of trapezium remains 
unchanged. In the definition of rhomboid two marks 
are lost, for he now omits to note that there are two 
obtuse angles and two acute angles. The total mark 
for this definition is 18. The definition of a diagonal 
of a square remains unchanged, and receives four 
marks as before. Two months after learning the 
definition F. G. receives 40 marks for his reproduc- 
tive test, against 44 marks in his test of immediate 
reproduction, and 42 marks in his first test of de- 
ferred reproduction, which took place one week after 
he learnt the definitions. He loses very little ; he had 



FIFTH SERIES OF EXPERIMENTS. 127 

evidently understood the definitions as well as learnt 
them. Indeed, his understanding is shown by his 
power of 'transfer,' for he receives a very good 
mark for his application to new material. 

Bearing in mind that this pupil worked in the de- 
ductive group, let us compare his work with the cor- 
responding papers of one of the best boys in the in- 
ductive group. 

George H , aged 13 years 9 months, in his test 

of immediate reproduction, wrote : 

1. A rhombus is a 'figure,' sides, four sides, all equal, four 
angles, two opposite sides are parallel, other sides are parrallel, 
all sides are straight. 

2. A trapezium is a figure, sides, of four, all unequal, four 
angles, angles unequal, all sides are straight. 

3. A rhomboid is a figure, of sides, four sides, two opposite 
sides are equal, parrallel, and has four angles, two opposite angles 
are equal, sides straight, two opposite sides straight, two sides are 
longer than the other pair of sides. 

4. A diagonal of a square is a line from one corner to the oppo- 
site corner, it is also straight. 

G. H.'s paper is, like F. G.'s, an excellent one. 
There is a certain staccato utterance which is a little 
irritating, but it is a peculiarity of his own and is not 
shared by the members of the inductive group gen- 
erally. For the definition of rhombus he receives 
single marks for 'figure,' 'four,' 'sides,' 'equal,' 
'straight,' 'four,' 'angle;' two marks for noting 
a pair of parallel sides, and that they are opposite 
sides ; and one mark for noting the other pair of par- 
allel sides; but he fails to note that the other pair of 
parallel sides are opposite also. He also receives 
two marks for noting that one pair of angles are 
equal and opposite. His total mark, therefore, for 
this definition is 12. His definition of trapezium re- 
ceives full marks. The definition of rhomboid is not 



128 INDUCTIVE VS. DEDUCTIVE METHODS. 

SO good. He receives single marks for 'figure,' 
'four,' 'sides,' 'straight,' 'four,' 'angles;' three 
marks for noting that one pair of opposite sides are 
'equal,' 'opposite' and parallel; two marks for not- 
ing that one pair of angles are equal and opposite, 
and two marks for stating that two sides are longer 
than the other two — a total of 13 marks. The defini- 
tion of diagonal scores full marks. G. H. thus re- 
ceives a total of 37 marks for his exercise in imme- 
diate reproduction. 

In a week's time, when he takes his first test in 
deferred reproduction, he obtains one mark less. In 
his definition of rhombus he omits the parallelism of 
the 'other sides,' losing a mark which he had gained 
the week previous. His definition of trapezium re- 
mains unchanged. In the definition of rhomboid, 
though it is expressed with some slight differences, 
he obtains all the marks which he received before, 
namely, 13. The definition of diagonal remains un- 
changed; for this he obtains four marks, as before, 
making a total of 36 marks. 

Seven weeks later there is a somewhat more seri- 
ous loss. He still receives 11 marks for the definition 
of rhombus, which has remained unchanged. His 
definition of trapezium has improved, for, though it 
contains no further units of correct description ac- 
cording to our scale, he notes that the smallest angle 
is opposite the smallest side and the biggest angle is 
opposite the biggest side. These statements are not 
quite clear, but indicate the commencement of a fresh 
idea about the trapezium. Two marks on his pre- 
vious record are lost in his definition of rhomboid; 
he omits now that there are two long and two short 
sides. The last definition remains unchanged. For 



FIFTH SERIES OF EXPERIMENTS. 129 

the second paper in deferred reproduction, there- 
fore, Gr. H. receives 34 marks. 

These papers written by F. G, and G. H,, though 
much superior to the average work, are typical in the 
slowness with which points like these of definitional 
description are forgotten when they have been duly 
understood, and expressed in a way which is really a 
result of work on the part of the pupil himself. 

5. The Test of Application to New Material. 

This, after all, is the supreme test of what teachers 
call 'intelligence.' We have seen in the two papers 
given above that the boy who learnt deductively knew 
more of what he had actually studied than the boy 
taught inductively, not only immediately after the 
work, but after two months had elapsed ; and with the 
boys of this class we shall find this difference to be 
true generally between the boys of the deductive and 
the boys of the inductive groups. The older children 
hitherto — girls and boys graded as Standard VII and 
upwards — have not shown this difference, though the 
younger and less proficient children have. I incline 
to attribute this to the relative i^redominance of de- 
ductive work in the usual teaching of this class, 
whereas in the two preceding classes of elder chil- 
dren, both boys and girls, the teaching was predomi- 
nantly inductive. Are we about to find that these 
boys give us results which differ from those of the 
older children previously experimented with, and, 
indeed, from all the children previously experi- 
mented with, when test is made of their power to 
apply their knowledge to new material? 

The test of application was the same as that used 



130 INDUCTIVE VS. DEDUCTIVE METHODS. 

with tlie Ex- VII class in the Higher Grade Boys' 
School. Drawings of hexagons, tangents to circles, 
pentagons and quadrilateral similar figures were 
shown. The questions: ''What is a hexagon?" 
"What is a tangent to a circle?" "What is a penta- 
gon?" and "In how many ways does ABCD resem- 
ble EFGH?" were written on the blackboard and the 
children answered them in writing.* 

I will illustrate what the boys did by means of two 
papers, both above the average, one from the 'de- 
ductive' and one from the 'inductive' group. 

Harry W., aged 13 years 6 months, who worked in 
the 'deductive' group, wrote: 

1. A hexagon is a figure with six straight sides all of which are 
equal, it has also six equal corners or angles. 

2. A tangent to a circle is a straight line, drawn so that it 
touches the circumference of the cii'cle. 

3. A pentagon is a figure with five straight sides and five angles, 
all sides being equal and all angles being equal. 

4. A. b. c. d. is the same as E. f. g. h. They vary by the sides, 
and the angles, if you look at them closely and then measure the 
angles they will all be different on one and all the same as the 
first on the other. They look different by the size. 

With the exception of the last definition, this is an 
easy paper to mark. The definition of hexagon re- 
ceives a total of 8 marks. The definition of tangent 
receives 3 marks. The definition of pentagon re- 
ceives 8 marks. In the last answer about the simi- 
larity of the quadrilateral figures, it is clear that H. 
W. wishes to express the inequality of the angles in 
both figures and the equality of the angles, each to 
each, of one figure with those of the other, for which 
he receives 5 marks. Thus H. W., taught deductively, 



♦The drawings may be seen ou page 108. One of the two similar 
quadrilaterals was lettered EFGH on this occasion. 



FIFTH SERIES OF EXPERIMENTS. 131 

scores 24 marks for his test of application to new 
material. 

Frank C , aged 13 years 2 months, who was 

taught inductively, wrote : 

1. A Hexagon is a six straight sided figure, liaving all sides 
equal, it has six angles equal, larger than right Angles. 

2. A Tangent to a circle is a line which is straight and is just 
touching the boundary of a circle. 

3. A Pentagon is a five, equal, straight sided figure, it has five 
equal angles larger than right angles. 

4. Both have four sides. 
Both have four angles. 

Both have four angles which are larger than right angles. 

A angle equals E angle. 

B " " F angle. 

C " " G angle. 

D " " II angle. 

Both have sides with the same slope. 

Both are placed on the same side. 

The definition of hexagon receives a total of nine 
marks; the definition of tangent three marks; and 
that of pentagon nine marks. The last answer is more 
difficult to mark. Both figures have ' sides ;' this car- 
ries one mark. There are four sides in both figures ; 
this carries another mark. Similarly, ''Both have 
four angles ' ' carries two marks. The next statement 
is wrong; it is not true that both have four angles 
which are larger than right angles. Then there are 
four marks for noting the equality of the angles, each 
to each, and four marks for noting that the sides of 
the figures are parallel. One further mark is gained 
by F. G.'s statement that both the figures are on the 
same side (of the base). This answer, therefore, re- 
ceives a total of 13 marks. The paper is an excellent 
one, and carries a total of 34 marks ; it is, in fact, one 
of the best papers worked in either group in the test 
of application to new material. 



132 INDUCTIVE VS. DEDUCTIVE METHODS. 

Lest the reader should carry away a quite exag- 
gerated notion of the power of application of these 
pupils (I am using the expression 'application' in 
the strictest sense), I propose to give one further 
paper by a boy who worked in the deductive group 
and made very little application of his knowledge. 
He obtained 37 marks in his test of immediate repro- 
duction and 34 marks a week after. But he obtained 
a very poor mark when he worked on new material, 
and seven weeks later he sank to 23 marks when 
tested on his old knowledge. There are evidently 
some boys who learn quickly and forget quickly. The 
pedagogical error, now happily being rectified by 
psychologists, has been to regard these boys as the 
rule rather than the exception. This boy, George L., 
aged 12 years 4 months, wrote : 

1. A Hexagon is a six sided figure. Each of the six sides are 
straight equal and opposite and Paralled. 

2. A tangent of a circle is a straight line dra\vn which is slant- 
ing and the circle stands on it. 

3. A Pentagon is a figiu'e with five sides, they are all straight. 
The Three small ones are equal and opposite, and the two long 
ones are equal and opposite. 

4. a. b. c. d's. has two straight long sides equal and the other 
tAvo sides unequal E. f. g. h's has two long straight sides equal, 
and the other two unequal only a. b. c. d is smaller than E. f. g. II. 

"Equal and opposite" has, unfortunately, trans- 
ferred itself too successfully. For his definition of 
hexagon he receives 5 marks. * * Each of the sides are 
opposite and parallel" is considered to be too con- 
fused to gain positive marks, but is not regarded as 
involving 'bad errors.' The definition of tangent 
gains 2 marks only; the latter part of his definition 
was drawn from one figure only. The statements 
that the tangent is slanting and that the circle stands 



FIFTH SERIES OF EXPERIMENTS. 133 

on it were not trne of all the tangents drawn, and are 
considered 'bad errors.' In Ins definition of penta- 
gon he receives 4 marks only. It is considered a 'bad 
error' to say that "three small ones are opposite." 
No positive marks are allowed for saying that "three 
are equal" and "two are equal," and it is counted an 
error to say there are "two long" and "three small" 
sides. In his last answer G. L. receives 2 marks; 
both the figures have 'sides,' and in each case two are 
longer than the remaining two. But none of them 
were equal; though, as two of them were not very 
different in length, the statement was not accounted 
a 'bad error.' The statement as to the size of the 
two figures is irrelevant; the boys were asked for 
'resemblances,' not for differences. This is one of 
the worst papers in the class. The boy had acquired 
the knowledge of the definitions of rhombus, etc., but 
he could not apply it, and he speedily forgot it. 

Possibly, with these examples before him, the 
reader may find greater interest in the tabulated 
results, which I now give. 

6. Results, 
(a) Of the Preliminary Tests. 

In the Preliminary Test the highest mark gained 
by any boy was 19, the lowest was 6, and the average 
mark was 12.25. There were 16 boys in the group 
deductively taught and 16 boys in the group induct- 
ively taught. The average mark of the boys of the 
first group was 12.25 (mean variation 3.0), and of 
those in the second group was 12.25 (mean variation 
3.0) . The correlation between the total results of the 
corresponding boys in the two groups was practi- 



134 INDUCTIVE VS. DEDUCTIVE METHODS. 

cally perfect, amounting to + .97 on the product- 
moment formula. In so far as one test can in any 
way be satisfactory as a basis of the division of a 
class into equal groups, it seems fair to suppose that 
an adequate division has been made. These boys, it 
will be remembered, had had some special inductive 
teaching concerning the square, triangle, etc., though 
I should not describe the general methods of their 
teacher as predominantly inductive. I incline to 
think this special inductive propaedeutic may have 
been an advantage to us in making the division, but 
it may, I fear, serve to throw some bias on the in- 
ductive side and unduly favor the inductive group. 
We may, however, remember that we have three ex- 
periments already described in which no such propae- 
deutic was given. 

(h) Of Immediate Reproduction. 

What marks did the two groups obtain immedi- 
ately after the teaching and learning? In two pre- 
vious experiments with older children, girls as well 
as boys, the group taught inductively appeared to 
advantage from the first. Is that also the case with 
these Standard VII boys I We can say quite defin- 
itely that it is not. The average mark obtained by 
the boys of the deductive group was 34.2 (mean vari- 
ation 6.0), and of the inductive group 31.4 (mean 
variation 4.5). 

This difference between the means and its prob- 
able error justify us statistically in asserting the 



FIFTH SERIES OF EXPERIMENTS. 



135 



existence of a general tendency in favor of the 'de- 
ductive' group. The superiority of the work of the 
deductive group in immediate reproduction may 
also be shown compendiously in the following table : 

Table XXVI, sJioivhig the tcork of the Deductive and Inductive 
Groups compared, section by section, in the Preliininaru Test 
and the Test of Immediate Reproduction. 





, Deductive Group. , 


, Inductive G 


roup. \ 








Av. mark 






Av. mark 






Average 


in imme- 




Average 


in imme- 






mark in 


diate 




mark in 


diate 


Marks in 


No. 


prelimi- 


repro- 


No. 


prelimi- 


repro- 


preliminary 


of 


nary 


duction 


of 


nary 


duction 


test. 


boys. 


test. 


test. 


boys. 


test. 


test. 


Over 15 


.. 4 


17.0 


38.0 


4 


17.0 


34.5 


10 to 15.... 


. . 7 


12.4 


32.7 


7 


12.5 


29.6 


5 to 10.... 


.. 5 


8.2 


33.2 


5 


8.0 


31.4 



(c) Correspondence Between Immediate and De- 
ferred Reproduction. 

But, after all, the important question in education 
is not so much what can be done by pupils immedi- 
ately after they have just been taught, but what they 
can do some time afterwards. Do they remember 
what they once knew, and how far can they apply 
their knowledge? To the second of these questions 
I hope to give an answer when dealing with the re- 
sults of the test on new material. Let me turn for a 
while to the first, and let me break it up into a number 
of constituent questions. The boys gain certain 
marks immediately after teaching and learning. 
What do they gain a week later, and, more important 
still, what do they gain two months later ? 



136 



INDUCTIVE VS. DEDUCTIVE METHODS. 



In a rough way we can find the answers to our 
questions in the following table : 

Table XXVII, slion-ing the work of the Inductive and Deductive 
Groups compared, section by section, in the Tests of Immediate 
and Deferred Reproduction. 

Deductive Group. 



Imme- 
diate 
Maries for No. repro- 

immediate of duetion 

reproduction. boys. test. 

40 and over 5 41.4 

35 to 40 3 37.7 

30 to 35 4 30.8 

25 to SO 3 26.7 

Below 25 1 22.0 

Inductive Group. 



-Average Marks. n 

Deferred Deferred 



Marks for No. 

immediate of 

reproduction. boys. 

40 and over 

35 to 40 6 

30 to 35 3 

25 to 30 G 

Below 25 1 



37.0 
32.0 
27.3 

20.0 



repro- 
duction, 
first 
test. 
38.6 
38.0 
28.3 
22.3 
20.0 



repro- 
duction, 
second 
test. 
38.4 
32.7 
25.8 
23.0 
19.0 



Imme- 
diate 
repro- 
duction 
test. 



-Average Marks.- 



Deferred 
repro- 
duction. 

first 

test. 

33!7 
29.3 

22.8 
17.0 



Deferred 
repro- 
duction, 
second 
test. 

3i!7 
27.0 
24.5 

24.0 



The conclusions seem clear. The Inductive Group 
contains no boys at all equal to the highest section 
of the Deductive Group. The best boys in the In- 
ductive Group correspond to the second section of 
the Deductive Group, but even then they are inferior 
to that section, both in the immediate and deferred 
tests. The work done in immediate reproduction 
may be very well taken as representative of what the 
work will be later on in exercises of this kind, for the 
various sections into which the groups are divided 



FIFTH SERIES OF EXPERIMENTS. 137 

retain their relative positions throughout the whole 
experiment. Calculated exactly, the correlation co- 
efficients between the results of Immediate Repro- 
duction and those of the first Deferred Reproduction 
Test in the Deductive Group is + .804, and between 
Immediate Reproduction and the second Deferred 
Reproduction Test (two months later) is + .859. 
The corresponding figures for the Inductive Group 
are + .616 and + .619. 

Summarizing the results and treating the groups 
as wholes, the averages and variabilities are as fol- 
low: 

Table XXVIII, showing the icorJc of the Inductive and Deductive 

Groups compared in the Tests of Immediate and Deferred 
Reproduction. 

Imme- First Second 

diate deferred deferred 

repro- repro- repro- 

Deductive Group : duction. duction. duction. 

Average mark 34.2 32.5 30.1 

M. V 6.0 6.3 7.0 

Inductive Group : 

Average mark 31.4 27.8 27.6 

M. V 4.5 5.1 4.1 

The Deductive Group has outdistanced the In- 
ductive Group quite clearly, both in immediate and 
deferred reproduction, not only in positive marks, 
for, perhaps, I ought to add, it has also made fewer 
'bad errors.' It is the third result in which this has 
been found to be the case. We shall, therefore, again 
have to admit the contention urged against induct- 
ive methods in the earlier chapters of this mono- 
graph. We must certainly conclude that, in exami- 
nations on precisely what has been taught or learnt, 
children taught by what we have called deductive 
methods may be more successful than children taught 



138 



INDUCTIVE VS. DEDUCTIVE METHODS. 



inductively. Also we see that children need not be 
young to be taught successfully by deductive meth- 
ods. Let us now turn, however, to the Test of Ap- 
plication to New Material and see whether the same 
relation between the two groups holds there. 



(d) Results of the Test on New Material. 

We have seen that for purposes of immediate, and 
even of deferred, reproduction the more mechanical 
method has shown itself superior to the less mechan- 
ical. Is the same relationship retained between the 
two groups when the test is no longer one of simple 
reproduction, but requires a transfer of knowledge 
or method to analogous material? We can say at 
once that the same relation is not maintained. The 
inductive group now comes to the front, but the dif- 
ference between the means of the two groups is a 
small one and the variability of the averages is high. 
The deductive group scores an average mark of 20.5 
(mean variation 5.9), and the inductive group an 
average mark of 21.1 (mean variation 4.4). But let 
us look a little more closely into the composition of 
these averages: 

Table XXIX, showing the worlc of the Inductive and Deductive 
Groups compared in Immediate Re product ion and in the Test 
on Neiv Material. 

, — Deductive Group. — n , Inductive Group. n 

Marks Average Maries for Average MarliS for 
in imme- Imme- Imme- 
diate No. diate No. diate 
repro- of repro- New of repro- New 
duction. boys, duction. material, bovs. duction. material. 

Over 35 8 40.0 2.S.0 37.0 24.8 

30 to 35 4 30.8 17.8 3 32.0 19.3 

25 to 30 3 20.7 19.7 (5 27.3 17.5 

Below 25 1 22.0 12.0 1 20.0 15.0 



FIFTH SERIES OF EXPERIMENTS. 139 

The figures certainly suggest a superiority on the 
side of the inductive group in three of the corre- 
sponding sections into which the groups are divided ; 
and the regiihir decline of the figures in both groups 
(with the exception of the average of 19.7 in the third 
section of the Deductive Group) would appear to in- 
dicate that there is a general tendency in favor of 
correlated transfer in the Inductive rather than in 
the Deductive Group. The coefficient of correlation 
between the results of the Inductive and Deductive 
Groups, when tested on new material, is, however, 
not very high. With high variability as well, this 
involves a high probable error. So that we may con- 
clude in this case merely that the Inductive Group 
does better work on the whole than the Deductive 
Group, but we have not the usual statistical justifi- 
cation that there is a strong general tendency in that 
direction. We shall, however, hardly feel disposed 
to attribute the superiority of the Inductive Group 
to chance, since in every one of the five experiments, 
with different teachers, with children of different 
ages, of different abilities and of different sexes, we 
have found the inductively taught group the more 
competent when tested on the power of application 
to new material. 



IX. GENERAL SUMMARY. 

In five different schools in different parts of Lon- 
don, attended by cliildren varying in social class, ex- 
periments have been made to test the relative values 
of 'inductive' and 'deductive' methods of teaching 
as applied to geometrical definition. Both girls and 
boys, of ages ranging from 8 to 15 years, were set to 
do the work. The main problems were two in num- 
ber. In the first place, an attempt was made to dis- 
cover which of the two methods gave the better re- 
sults when the children were tested on precisely what 
they had been taught or had learnt. In the second 
place, an endeavor was made to find out which of the 
two methods gave the better results when the chil- 
dren were tested on new material. 

The answer to the first of these two questions was 
not the same in all of the five schools tested. In three 
of them, two of the three boys' schools and one of the 
two girls' schools, the conclusion was unambiguously 
in favor of the 'deductive and memoriter' method. 
This was the case with the younger and less profi- 
cient boys and girls, and at first sight it looked as if 
age were an important factor in the ]:>roduction of 
this result, but the same result was obtained with a 
class of boys who were much older, so that age was 
certainly not the only factor of differentiation. In 
two classes, the oldest class of boys and the oldest 
class of girls who did the work, the inductive method 

140 



GENERAL SUMMAEY. 141 

was just as successful as the 'deductive,' even for 
purjDoses of exact reproduction, immediately after- 
wards, of what had been taught or learnt. There 
were some indications that the children inductively 
taught lost rather less of what they had known than 
those deductively taught when they were tested some 
time afterwards; but, on the whole, the tests of de- 
ferred reproduction gave the same comparative re- 
sults as those of immediate reproduction. The im- 
portance of this consideration in testing school 
methods where exact reproduction is required is 
obvious.. 

The answer to the second of the two main issues 
was the same in all of the five schools tested. The 
children who were taught 'inductively' did better 
work than those taught 'deductively' in every case 
when they were required to apply themselves to new 
material. 

This research, therefore, offers an experimental 
justification of what are known, among teachers, as 
' intelligent ' methods of teaching, and of the superior 
'transfer' effect of certain methods. 

Many pedagogical corollaries may be drawn from 
the experiments, but it will be sufficient in this place 
to emphasize a consideration already alluded to in 
the body of the text. 

Examinations, whether internal, that is, conducted 
from within by the school authorities, or external, 
that is, conducted by external educational authori- 
ties, should always include questions on subject-mat- 
ter which is not identical with that set down in the 
syllabuses of instruction if the examination is to test 
good method in teaching. But if the tests are to 
serve any useful pedagogical purpose, the new mate- 



142 INDUCTIVE VS. DEDUCTIVE METHODS. 

rial, though it should not be identical, ought to be 
analogous to that which has been dealt with in the 
school curriculum. Questions on new analogous ma- 
terial are probably the best questions of all (if the 
same set of questions be required to serve a double 
purpose), for they test, with fair adequacy, whether 
the work set down in the syllabuses has been effi- 
ciently done, and they also test, with admirable ade- 
quacy, whether the methods by which the school work 
has been done were such as to give the pupil power 
to apply his knowledge. 



INDEX 

'Bad' errors, meaning of, 36. 

and mechanical method, 51. 
Chance or Variability, 7. 

Children's Definitions, spontaneous, 27, 28, 57, 71, 72, 73, 
74, 102, 103, 120, 121. 

after teaching and learning, 76, 77, 78, 80, 81, 82, 125, 
127. 

of new analagous material, 40, 83, 84, 85, 86, 88, 90, 
109, 111, 130, 131, 132. 
Circle, definition of diameter of, 29, 32. 

drawing of diameter of, 24. 
Classes taking the experiment, 20, 23, 55, 69, 100, 119. 
Co-operation of Teachers, 4. 
Correlation coefficients, 9, 10, 50, 62, 66, 91, 96, 115, 137. 

value of, 30. 
Deductive Method, method of learning by, 37. "*J^ 

method of testing, 19. 

shown to be the better, 46, 93, 134, 136, 137. 
Deferred Reproduction, 44, 47, 63, 65, 92, 107, 124, 135. 
Definitions, 'real,' 26, 27. 

arguments in favor of deductive treatment of, 17. 

arguments in favor of inductive treatment of, 18. 

units of marking of, 28, 29, 41, 42, 43. 

as learnt deductively, 31.:. \ 

as learnt inductively, 32. 

children's spontaneous, after teaching and learning, 
of new analagous material, see Children's Defini- 
tions. 

of diameter of circle, 29, 32. 

of hexagon, 113. 

of oblong, 29, 32. 

143 



144 INDUCTIVE VS. DEDUCTIVE METHODS. 

of pentagon, 113. 

of rhomboid, 42, 106, 123. 

of rhombus, 41, 105, 123. 

of square, 25, 28, 32, 35. 

of diagonal of square, 39, 106, 124. 

of tangent to circle, 113. 

of trapezium, 42, 106, 123. 

of triangle, 28, 32. 
Demonstrative Geometry, introduction to, 17, 27. 
Diagonals of Squares, 39, 106, 124. 
Diameter of Circle, definition of, 29, 32. 

drawing of, 24. 
Durability of knowledge, 19, 37, 48, 49, 64, 66, 95. 
Education, method of settling disputed questions in, 16. 
Educational Science, 3. 
Equal groups, how formed, 30, 45, 56, 61, 92, 133. 

use of, 52. 
Errors, 'bad,' meaning of, 36. 

and mechanical method, 51. ^ 

Errors, method of correcting inductively, 34. 'X 

'probable errors,' method of determining, 8-10. 

in spelling not counted, 43. 
Experiment, use of, 16. 
Experimental Pedagogy, 1. 
Geometrical Definitions, see Definitions, 
Geometrical Teaching, alleged cause of 'chaos in,' 18. 
Geometry, Demonstrative, introduction to, 17, 27. r^. 
Groups, equal, how formed, 30, 45, 56, 61, 92, 133. 

use of, 52. 
Hexagon, definition of, 113. 

drawings of, 108. 
Immediate Keproduction, 19, 44, 47, 63, 65, 92, 107, 124, 

135. 
Inductive Method, an objection to, 14. "y" 



; INDUCTIVE VS. DEDUCTIVE METHODS. 145 

arguments for, 18. ^ 

method of learning by, 32. ^ 

method of correcting by, 34. 

method of testing, 19. 

shown to be the better, 51, 52, 61, G(3, 67, 98, 116, 117, 
138. 
Intelligence, meaning of, 18. 

test of, 38, 129. 

training of, 53. 
Knowledge, durability of, 19. 

relation between quickness and permanence, 37, 48, 
49, 64, 66, 95. 
Marks, positive and negative, 36, 38. 
Material, 'new,' see 'New Material.' 
Negative marks, 36, 38. 
'New material,' meaning of, 53, 54. 

application to, 19, 38, 50, 67, 96, 107, 116, 129, 138. 
'New methods,' general tendency of, 13, 14. 

alleged disadvantages of, 14. 
Novelty, influence of, 70. 
Oblong, definition of, 29, 32. 

drawings of, 24, 
Pedagogy, Experimental, 1. 
Pentagon, definition of, 113. 

drawings of, 108. 
Positive Marks, 36, 38. 
Practice versus Theory, 11, 13, 15. 
'Probable Errors,' method of determining, 8-10. 
Reproduction, deferred, 44, 47, 63, 65, 92, 107, 124, 135. 

immediate, 19, 44, 47, 63, 65, 92, 107, 124, 135. 
Rhomboid, definition of, 42, 106, 123. 

drawings of, 39. 
Rhombus, definition of, 41, 105, 123. 

drawings of, 39. 



146 INDUCTIVE VS. DEDUCTIVE METHODS. 

School Classes taking the experiment, 20, 23, 55, 69, 100, 

119. 
Science, Educational, 3. 
'Science' of Education, 4. 
Spelling errors not counted, 43. 
Spontaneous definitions, 27, 28, 57, 71, 72, 73, 74, 102, 103, 

120, 121. 
Square, definition of, 25, 28, 32, 35. 

diagonals of, 39, 106, 124. 

drawings of, 24. 
Tangent to Circle, definition of, 113. 

drawings of, 108. 
Teaching, the divergence of Theory and Practice, 11. 

breach between Theory and Practice, 13, 15. 

unintelligent, reaction against, 70. 
Teachers, co-operation of, 4. 
Theory versus Practice, 11, 13, 15. 
Time taken for the exercises, 44, 59, 75, 104, 106, 122. 
Trapezium, definition of, 42, 106, 123. 

drawings of, 139. 
Unintelligent teaching, reaction against, 70. 
Unsophisticated material, 20. 
Variability or Chance, 7. 



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Moto- 
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Develop- 
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Observations 
on the First 
Three Years 
of a Child. 



By 
GEORGE 
V. N. 
SEARBOBN 



Price: 

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Few subjects are of greater interest to 
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To the psychologist, and to a less extent 
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Spelling 

Efficiency 

in 

Relation 

to Age, 

Grade 

and Sex, 

and the 

Question 

of 

Transfer 



An Experi- 
mental and 
Critical 
Study of the 
Function of 
Metliod in tlie 
Teacliing of 
Spelling. 



By 
J. E. 
WAIiIiACE 



Price: 

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vi, 91 pages. 

$L26. 



There are few elementary school sub- 
jects in whicli inefBciency is more surely 
detected and reprobated in later life, and 
in the teaching of which the elementary 
schools are charged with more extrava- 
gant waste of time, than spelling. 7.22 
per cent, of the time of the child in the 
elementary schools in ten of our largest 
cities is devoted to the study of spelling, 
and yet the complaint continues to be 
almost universally voiced that the ele- 
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have not learned how to spell. 

School superintendents and teachers 
have felt the justice and sting of these 
criticisms, and have attempted to pro- 
vide a remedy either by increasing the 
time devoted to spelling or by changing 
the methods of teaching. The results, 
however, have not in all cases proved 
satisfactory. 

Dr. Wallin, who has been offering 
courses in educational psychology and the 
principles of teaching in schools of edu- 
cation for a number of years, points out 
briefly in this monograph some of the 
fallacies involved in the exclusive use of 
the incidental method of teaching spell- 
ing, based upon the psychological prin- 
ciples which condition the reduction of 
mechanical subject-matter to the plane of 
automatism (spelling is of an instru- 
mental nature). By means of the re- 
sults of the very researches made in the 
past to demonstrate the adequacy of the 
incidental method, it is shown that its 
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periority of a spelling drill technique, 
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When 
Should a 
Child 
Begin 
School ? 

An Inquiry 
Into the 
Relation 
Between the 
Age of Entry 
and School 
Progress. 



By 

vr. H. 

WINCH 



Few educational questions have excited 
more general interest in recent years 
than that of the age at which children 
suould commence their attendance at 
school. On the one side we have the 
rule-of-three conclusion, felt rather than 
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children of Great Britain do. While this 
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standing than entry at a later age? In 
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pupils who entered school earlier found 
to constitute the younger portion of the 
class? 2. In the same grade some pupils 
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by examinations? 3. How far does early 
entry depend upon social circumstances? 
4. What is the influence of early entry 
upon the subsequent behavior of pupils 
and upon their attentiveness to school 
work ? 

The results of Mr. Winch's inquiry are 
now published for the first time. Some 
Vrti'f °^ them have been privately circulated, 

'^'^"^''- and a few of the tables, together with 

12mo cloih, the methods employed, were discussed 
1AC '«/.«». some years ago at a meeting of the In- 

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Mental 
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"Die 

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ErmUdung.' 



By 

Translated 
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German by 
GUT 

BIONTBOSZ; 
WHZFFI.B 



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This noteworthy monograph ia a com- 
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delivered Ix^fore the Munich association 
of gymnaslal teachers, and its primary 
purpose is not to contribute to the ex- 
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to inform and to interest teachers. 

The following are amoog the topics dis- 
cussed : The nature and forms of fatigue, 
the symptoms of fatigue, the measure- 
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psychological methods, the factors other 
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tal work— practice, adaptation, warming- 
up, spurts, enthusiasm, etc. — and the 
laws of fatigue. 

In considering the application of these 
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is given to the dependence of fatigue 
upon individual differences, upon age, 
puberty, the length of lesson periods, the 
number of lessons per day, the day of 
the week, the introduction of various 
rest pauses (recesses, holidays, vacations, 
etc.), change of occupation, the fatigue 
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The translation is offered with the con- 
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educational psychology and of the hy- 
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mental fatigue and its relation to school 
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Alpha- 
bets 



An Experi- 
mental Inves- 
tigation of 
the Compara- 
tive Merits of 
tlie Webster 
Key Alphabet 
and the 
Proposed Key 
Alphabet 
Submitted to 
the National 
Education 
Association. 



By 
GUY 

MONTBOSZ: 
WHIFPIiE, 



Price: 

Svo, 60 pages 

35c. paper 

binding. 

WARWICK 



This monograph will exert a two-fold 
appeal to those who aim to keep abreast 
of present-day movements in education. 
First, in that it offers an excellent ex- 
ample of the application of the experi- 
mental method to a pedagogical problem, 
and in this respect will take its place as 
a contribution to experimental pedagogy ; 
secondly, in that it deals with an im- 
portant topic just now a matter of gen- 
eral discussion in educational circles. 

The National Education Association 
has under consideration the adoption of 
a new key-alphabet for phonetic nota- 
tion. The merits of the proposed alpha- 
bet have been the subject of extensive 
and lively debate, but no one has hither- 
to done the obvious thing and tried out 
the new alphabet under experimental 
conditions. This Dr. Whipple has ac- 
complished, and the results will interest 
every teacher who uses a phonetic alpha- 
bet in his class work as well as every 
educator who believes with the author 
that, in the school as well as in other 
realms of life, "you can tell by trying." 

In view of the fact that the subject of 
phonetic alphabets will be given much 
attention by educators during the next 
year, this work is offered at a price 
which will place it easily in reach of 
teachers in city and rural schools, and 
also the members of clubs and reading 
circles. 

& VOSX, Inc., BAI.TXMOBZ:, MD. 



Back- 
ward and 
Feeble- 
Minded 
Children 

A Series of 
Studies 1b 
Clinical 
Psychology. 



By 
EDMUNB 
B. SXTBT 



Price: 

12m 0, 

200 pages, 

illut. 

$1.40. 



Bach of the more populous States has 
several thousand mental defectives, large 
numbers of whom are attending the put>- 
lic schools. They usually make little 
progress and are distressingly disturbing 
factors In the regular classes. In Ger- 
many, and recently in Prance, and in 
some of our own cities, these children 
are being placed in special classes or in 
special schools, according to the degree 
of defect. Teachers and school experi- 
ence immediate relief, and the children 
themselves are the greatest beneficiaries. 
All the schools have these defectives, and 
the problem of recognizing and caring 
for them is an immediately pressing one 
In all our cities, towns and rural dis- 
tricts. 

Following a yenx in the clinics of Paris, 
Dr. Huey's posit on at Lincoln for nearly 
a year and a half Involved making a 
mental examination of each new ad- 
mission to this, one of the largest state 
institutions for the feeble-minded. 

As research psychologist to the insti- 
tution Dr. Huey made careful psychologi- 
cal study of 35 selected cases which rep- 
resent the transition zone between feeble- 
mindedness and non-feeble-mindednesa. 
These are Just the border cases that puz- 
zle the school principal or the clinician. 
In this volume he presents case after 
case representing various types and 
groups of backward and feeble-minded 
children. The mental and physical char- 
acteristics of each child and the salient 
features of different groups are clearly 
stated, with charts which graphically 
present the results of various measure- 
ments and tests. 

The methods of making examinations 
and tests and of making observations and 
gathering data needed for the interpre- 
tation of any given case are illustrated 
in detail. The concreteness of the ma- 
terial and the abundance of illustrative 
examples will be appreciated by all, and 
make the studies intelligible even to 
those unfamiliar with psychological 
technique. 



WARWICK & YORK, Xnc, BAI^TIlllIORE, MD. 



Experi- 
mental 
Studies 
of Mental 
Defectives 

▲ Critique of 
the Binet- 
Slmon Tests 
and a Contri- 
bution to the 
Psycliology 
of Epilepsy. 



By 

J. E. 

WAI^ULCE 
WAI.I.XN, 
FhJ>. 



About 

160 pages. 

$1.25. 



The Binet-Simon tests liave be«D bailed 
by popular writers and even by some 
scientific workers as a wonderful mental 
X-ray machine, which will enable us to 
dissect the mental and moral mechan- 
isms of any normal or abnormal indi- 
vidual. But those who have had ex- 
tensive experience with these tests linow 
that, despite their very great practical 
value, they have numerous imperfections 
and definite limitations. These imperfec- 
tions and limitations can be made known 
only by thoroughgoing trial on large 
groups of individuals by expert investi- 
gators. Dr. Wallin is well qualified by 
training and experience to undertake this 
work, and he has presented in this, the 
seventh of the series of Educational 
Psychology Monographs, a systematic 
critical study of the results of the Binet 
Scale when applied to a colony of epi- 
leptic children, and has included a guide 
for the conduct of the tests. 

In the course of his study certain facts 
have been revealed concerning the men- 
tal status of the epileptic which should 
interest the schoolman as well as the 
alienist and the physician, for epileptic 
children constitute a numerous class 
which grades nearer the public school 
laggard than do feeble-minded children, 
and which cannot be reached by the cut- 
and-dried methods of the schools, but re- 
quires a special educational regime. 
Moreover, epilepsy, despite the investiga- 
tions of many alienists, still remains a 
little understood pathological condition 
with marked disturbance of mentality. 

We commend this contribution to the 
attention of physicians, alienists and all 
schoolmen who are interested in the 
scientific examination of mental de- 
ficiency. 



WARWICK & YOBK, Inc., BAIiTmOBE, MD. 



Varia- 
tions in 
the 

Grades of 
High- 
School 
Pupils 



By 
CIiABENCZ: 
TBUMAN 
GBAV. 



12mo, 
Cloth ca, 
120 pages. 

$1.25. 



Ten years ago no serious attempt had 
been made to study scientifically the 
relative merits of various systems of 
grading students, despite the fact that 
statistical methods for undertaking such 
studies were fully available and that 
grading plays so large a rOle in the 
school career of hundreds of thousands 
of school children. In the last five 
years, however, this inviting field has 
been the scene of numerous important 
investigations, so that we have at least 
arrived at a better understanding of the 
nature of the problem and of the general 
line along which progress must be made. 

In the present monograph Mr. Gray 
reports the methods and results of his 
investigation of one phase of the general 
problem, viz., the nature, degree and 
causes of the variations occurring in the 
grades of high-school pupils. The gen- 
eral aim of his study is to base an edu- 
cational investigation upon school grades. 
It is usually argued that such marks 
are inaccurate, that they are complex, 
that they are not scientific, and, above 
all, that it is impossible to measure 
mental traits by such cold statistics as 
grades afford. In direct contrast to 
these arguments stands the fact that 
all promotions from the kindergarten 
through the university are based upon 
this so-called inaccurate, complex, unsci- 
entific and cold estimates of progress 
and achievement. One of the most vital 
and fundamental principles of any school 
system is its plan of promotions, and 
because of the close relation between 
promotions and grades there is the most 
urgent need that schoolmen become in- 
terested in the problems of grading. 
Variations in the Grades of Hiffh-School 
Pupils should interest all teachers, and 
more particularly all school administra- 
tors, because the author not only shows 
clearly how unreliable are the grades 
commonly given by teachers, and makes 
evident the need of Instruction and train- 
ing in grading, but also presents a rela- 
tivelv simple method by means of which 
any "high-school principal can study the 
condition of the grading in his own 
school and take due steps to remedy the 
faults that he may find. 



WARWICK & YOBK, Inc., BAIiTIMGBZ;, MD. 



How I 

Kept My 

Baby 

Well 



By 
ANNA a. 
NOVES. 



i2moy 

Cloth, 

Illustrated, 

ca, 180 pages. 

$1.25. 

WABWICl 



The fact that the Journal of Educa- 
tional Psychology has defined its scope 
to include the consideration of child psy- 
chology and hygiene justifies the inclu- 
sion in the allied series of Educational 
Psychology Monographs of the material 
set forth in the present volume. 

Mrs. Noyes has made a contribution of 
real interest to physicians and nurses, to 
mothers and fathers, and to students of 
childhood generally. The value of her 
work is twofold. On the one hand, it 
points the way to a method and type of 
observation tliat any intelligent mother 
can undertake with profit to herself and 
to others, and in so far disproves the 
contention of some critics of the child- 
study movement that observations of 
young children by their own mothers can 
never yield data of real value ; on the 
other hand, it furnishes generalizations 
in the shape of principles or rules gov- 
erning feeding, clothing and the general 
control of infant development that will 
be of direct utility to those who, like 
the author, face that vital problem- 
how to keep the baby well. Mrs. Noyes 
has displayed commendable caution in 
drawing these generalizations. It is not 
asserted that what applied to her own 
baliy will apply invariably to any other 
baby, but only that it undoubtedly will 
apply to many babies, and that her 
method of attacking the problem is, at 
any rate, a method that other mothers 
may follow to advantage when confront- 
ed with the same situation. 

The conservation of human life by the 
reduction of infant mortality is a noble 
undertaking, and it is hoped that this 
little contribution may in some measure 
further that undertaking. 

The volume is profusely illustrated. 
The author and Mr. Noyes followed the 
life of the child through his first two 
years with a camera just as faithfully 
as the mother followed him with her 
charts and memorandum pad. As a con- 
sequence there appear as illustrations 
more than sixty pictures of the baby, 
most of them full-page cuts. The book 
also contains some forty or fifty full- 
page charts. Both photographs and 
charts greatly enhance the value of the 
book. 
&i YOBX, Inc., BAI^TZMOBi:, MD. 



Al^G 5 1913 



LIBRARY OF CONGRESS 



021 344 895 A 



